1

1 0 71115 478436 478434 2020-07-16T10:43:54Z Prime.mover 59 wikitext text/x-wiki {{NumberPageLink|prev = 0|next = 2}} == Number == $1$ ('''one''') is: :The immediate successor element of zero in the set of natural numbers $\N$ :The only (strictly) positive integer which is neither prime nor composite :The only (strictly) positive integer which is a divisor of every integer === $0$th Term === :The $0$th (zeroth) power of every non-non-zero number: ::$\forall n: n \ne 0 \implies n^1 = 1$ :The $0$th term of Göbel's sequence, by definition :The $0$th term of the $3$-Göbel sequence, by definition :The $0$th and $1$st Catalan numbers: ::$1 = \dfrac 1 {0 + 1} \dbinom {2 \times 0} 1 = \dfrac 1 1 \times 1$ ::$1 = \dfrac 1 {1 + 1} \dbinom {2 \times 1} 1 = \dfrac 1 2 \times 2$ :The $0$th and $1$st Bell numbers === $1$st Term === :The $1$st (strictly) positive integer :The $1$st (strictly) positive integer :The $1$st (positive) odd number ::$1 = 0 \times 2 + 1$ :The $1$st number to be both square and triangular: ::$1 = 1^2 = \dfrac {1 \times \paren {1 + 1}} 2$ :The $1$st square number to be the $\sigma$ (sigma) value of some (strictly) positive integer: ::$1 = \map \sigma 1$ :The $1$st generalized pentagonal number: ::$1 = \dfrac {1 \paren {3 \times 1 - 1} } 2$ :The $1$st highly composite number: ::$\map \tau 1 = 1$ :The $1$st special highly composite number :The $1$st highly abundant number: ::$\map \sigma 1 = 1$ :The $1$st superabundant number: ::$\dfrac {\map \sigma 1} 1 = \dfrac 1 1 = 1$ :The $1$st almost perfect number: ::$\map \sigma 1 = 1 = 2 - 1$ :The $1$st factorial: ::$1 = 1!$ :The $1$st superfactorial: ::$1 = 1\$ = 1!$ :The $1$st Lucas number after the zeroth $(2)$ :The $1$st Ulam number :The $1$st (strictly) positive integer which cannot be expressed as the sum of exactly $5$ non-zero squares :The $1$st of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes :The $1$st of the $5$ known powers of $2$ whose digits are also all powers of $2$ :The $1$st factorion base $10$: ::$1 = 1!$ :The $1$st of the trivial $1$-digit pluperfect digital invariants: ::$1^1 = 1$ :The $1$st of the $1$st pair of consecutive integers whose product is a primorial: ::$1 \times 2 = 2 = 2 \#$ :The $1$st of the (trivial $1$-digit) Zuckerman numbers: ::$1 = 1 \times 1$ :The $1$st of the (trivial $1$-digit) harshad numbers: ::$1 = 1 \times 1$ :The $1$st positive integer whose cube is palindromic (in this case trivially): ::$1^3 = 1$ :The $1$st lucky number :The $1$st palindromic lucky number :The $1$st Stern number :The $1$st Cullen number: ::$1 = 0 \times 2^0 + 1$ :The $1$st number whose $\sigma$ value is square: ::$\map \sigma 1 = 1 = 1^2$ :The $1$st positive integer after $1$ of which the product of its Euler $\phi$ function and its $\tau$ function equals its $\sigma$ function: ::$\map \phi 1 \map \tau 1 = 1 \times 1 = 1 = \map \sigma 1$ :The $1$st positive integer solution to $\map \phi n = \map \phi {n + 1}$: ::$\map \phi 1 = 1 = \map \phi 2$ :The $1$st element of the Fermat set :The $1$st integer $n$ with the property that $\map \tau n \divides \map \phi n \divides \map \sigma n$: ::$\map \tau 1 = 1$, $\map \phi 1 = 1$, $\map \sigma 1 = 1$ :The $1$st Lucas number which is also triangular :The $1$st tetrahedral number: ::$1 = \dfrac {1 \left({1 + 1}\right) \left({1 + 2}\right)} 6$ :The $1$st of the $3$ tetrahedral numbers which are also square :The $1$st trimorphic number: ::$1^3 = \mathbf 1$ :The $1$st powerful number (vacuously) :The $1$st integer which equals the number of digits in its factorial: ::$1! = 1$ :which has $1$ digit :The $1$st power of $2$ which is the sum of distinct powers of $3$: ::$1 = 2^0 = 3^0$ :The $1$st square which has no more than $2$ distinct digits :The $1$st pentagonal number: ::$1 = \dfrac {1 \left({1 \times 3 - 1}\right)} 2$ :The $1$st pentagonal number which is also palindromic: ::$1 = \dfrac {1 \left({1 \times 3 - 1}\right)} 2$ :The $1$st square pyramidal number: ::$1 = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6$ :The $1$st pentatope number: ::$1 = \dfrac {1 \paren {1 + 1} \paren {1 + 2} \paren {1 + 3} } {24}$ :The $1$st automorphic number: ::$1^2 = \mathbf 1$ :The $1$st number such that $2 n^2 - 1$ is square: ::$2 \times 1^2 - 1 = 2 \times 1 - 1 = 1 = 1^2$ {{WIP|Find out what sequence this belongs to, because it has not been set up on the $5$ page}} :The $1$st Ore number: ::$\dfrac {1 \times \map \tau 1} {\map \sigma 1} = 1$ :and the $1$st whose divisors also have an arithmetic mean which is an integer: ::$\dfrac {\map \sigma 1} {\map \tau 1} = 1$ :The $1$st hexagonal number: ::$1 = 1 \paren {2 \times 1 - 1}$ :The $1$st pentagonal pyramidal number: ::$1 = \dfrac {1^2 \paren {1 + 1} } 2$ :The $1$st heptagonal number: ::$1 = \dfrac {1 \paren {5 \times 1 - 3} } 2$ :The $1$st centered hexagonal number: ::$1 = 1^3 - 0^3$ :The $1$st hexagonal pyramidal number: :The $1$st Woodall number: ::$1 = 1 \times 2^1 - 1$ :The $1$st happy number: ::$1 \to 1^2 = 1$ :The $1$st positive integer the sum of whose divisors is a cube: ::$\map \sigma 1 = 1 = 1^3$ :The $1$st cube number: ::$1 = 1^3$ :The $1$st of the only two cubic Fibonacci numbers :The $1$st octagonal number: ::$1 = 1 \paren {3 \times 1 - 2}$ :The $1$st heptagonal pyramidal number: ::$1 = \dfrac {1 \paren {1 + 1} \paren {5 \times 1 - 2} } 6$ :The $1$st Kaprekar triple: ::$1^3 = 1 \to 0 + 0 + 1 = 1$ :The $1$st palindromic cube: ::$1 = 1^3$ :The $1$st Kaprekar number: ::$1^2 = 01 \to 0 + 1 = 1$ :The $1$st number whose square has a $\sigma$ value which is itself square: ::$\map \sigma 1 = 1 = 1^2$ :The $1$st of the $5$ tetrahedral numbers which are also triangular :The $1$st positive integer which cannot be expressed as the sum of a square and a prime :The $1$st positive integer such that all smaller positive integers coprime to it are prime :The (trivial) $1$st repunit :The $1$st fourth power: ::$1 = 1 \times 1 \times 1 \times 1$ :The $1$st integer $m$ whose cube can be expressed (trivially) as the sum of $m$ consecutive squares: ::$1^3 = \displaystyle \sum_{k \mathop = 1}^1 \left({0 + k}\right)^2$ :The $1$st and $2$nd Fibonacci numbers after the zeroth ($0$): ::$1 = 0 + 1$ :The $1$st positive integer whose $\sigma$ value of its Euler $\phi$ value equals its $\sigma$ value: ::$\map \sigma {\map \phi 1} = \map \sigma 1 = 1 = \map \sigma 1$ :The $1$st square pyramorphic number: ::$1 = \displaystyle \sum_{k \mathop = 1}^1 k^2 = \dfrac {1 \paren {1 + 1} \paren {2 \times 1 + 1} } 6$ :The $1$st of the $4$ square pyramidal numbers which are also triangular :The $1$st Wonderful Demlo number :The $1$st obstinate number :The index of the $1$st Cullen prime: ::$1 \times 2^1 + 1 = 3$ :The index of the $1$st Mersenne number which {{AuthorRef|Marin Mersenne}} asserted to be prime ::($1$ itself was classified as a prime number in those days) :The number of different representations of $1$ as the sum of $1$ unit fractions (degenerate case) :The $1$st centered hexagonal number which is also square :The $1$st pentagonal number which is also triangular: ::$1 = \dfrac {1 \paren {3 \times 1 - 1} } 2 = \dfrac {1 \times \paren {1 + 1} } 2$ :The $1$st odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime :The $1$st odd number which cannot be expressed as the sum of an integer power and a prime number :The number of distinct free monominoes === $2$nd Term === :The $2$nd after $0$ of the $5$ Fibonacci numbers which are also triangular :The $2$nd palindromic triangular number after $0$ :The $2$nd integer $n$ after $0$ such that $2^n$ contains no zero in its decimal representation: ::$2^1 = 2$ :The $2$nd integer $n$ after $0$ such that $5^n$ contains no zero in its decimal representation: ::$5^1 = 5$ :The $2$nd integer $n$ after $0$ such that both $2^n$ and $5^n$ have no zeroes: ::$2^1 = 2, 5^1 = 5$ :The $2$nd palindromic integer which is the index of a palindromic triangular number after $0$: ::$T_1 = 1$ :The $1$st palindromic integer after $0$ whose square is also palindromic integer ::$1^2 = 1$ :The $2$nd Dudeney number after $0$: ::$1^3 = 1$ :The $2$nd number after $0$ which is (trivially) the sum of the increasing powers of its digits taken in order: ::$1^1 = 1$ :The $2$nd non-negative integer $n$ after $0$ such that the Fibonacci number $F_n$ ends in $n$ :The $2$nd after $0$ of the $3$ Fibonacci numbers which equals its index :The $2$nd subfactorial after $0$: ::$1 = 2! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} }$ :The $2$nd integer $m$ after $0$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$: ::$1^2 = \dbinom 0 0 + \dbinom 0 1 + \dbinom 0 2 + \dbinom 0 3$ :The $2$nd integer $m$ after $0$ such that $m! + 1$ (its factorial plus $1$) is prime: ::$1! + 1 = 1 + 1 = 2$ :The $2$nd integer after $0$ such that its  double factorial plus $1$ is prime: ::$1!! + 1 = 2$ :The $2$nd integer after $0$ which is palindromic in both decimal and binary: ::$1_{10} = 1_2$ :The $2$nd integer after $0$ which is palindromic in both decimal and ternary: ::$1_{10} = 1_3$ :The $2$nd Ramanujan-Nagell number after $0$: ::$1 = 2^1 - 1 = \dfrac {1 \left({1 + 1}\right)} 2$ :The number of different representations of $1$ as the sum of $2$ unit fractions: ::$1 = \dfrac 1 2 + \dfrac 1 2$ :The number of distinct free dominoes === Miscellaneous === :The total of all the entries in the trivial magic square of order $1$: ::$1 = \displaystyle \sum_{k \mathop = 1}^{1^2} k = \dfrac {1^2 \paren {1^2 + 1} } 2$ :The total of all the entries in the trivial magic cube of order $1$: ::$1 = \displaystyle \sum_{k \mathop = 1}^{1^3} k = \dfrac {1^3 \paren {1^3 + 1} } 2$ :The magic constant of the trivial magic square of order $1$: ::$1 = \displaystyle \dfrac 1 1 \sum_{k \mathop = 1}^{1^2} k = \dfrac {1 \paren {1^2 + 1} } 2$ :The magic constant of the trivial magic cube of order $1$: ::$1 = \displaystyle \dfrac 1 {1^2} \sum_{k \mathop = 1}^{1^3} k = \dfrac {1 \paren {1^3 + 1} } 2$ {{ArithmeticFunctionTable|n = 1|tau = 1|phi = 1|sigma = 1}} == Also see == * Definition:Unity * One is not Prime * Divisors of One * One Divides all Integers === Previous in sequence: $0$ === ==== Next in sequence: $2$ ==== * {{NumberPageLink|prev = 0|next = 2|type = Subfactorial|cat = Subfactorials}} * {{NumberPageLink|prev = 0|next = 2|result = Sequence of Integers whose Factorial plus 1 is Prime}} * {{NumberPageLink|prev = 0|next = 2|result = Prime Values of Double Factorial plus 1}} * {{NumberPageLink|prev = 0|next = 2|type = Fibonacci Number|cat = Fibonacci Numbers}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 5 with no Zero in Decimal Representation}} * {{NumberPageLink|prev = 0|next = 2|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|prev = 0|next = 2|result = Palindromes in Base 10 and Base 3}} * {{NumberPageLink|prev = 0|next = 2|result = Numbers which are Sum of Increasing Powers of Digits}} * {{NumberPageLink|prev = 0|next = 2|result = Square Formed from Sum of 4 Consecutive Binomial Coefficients}} * {{NumberPageLink|prev = 0|next = 2|result = Palindromic Indices of Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 2|result = Square of Small-Digit Palindromic Number is Palindromic}} ==== Next in sequence: $3$ ==== * {{NumberPageLink|prev = 0|next = 3|type = Triangular Number|cat = Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 3|result = Palindromes in Base 10 and Base 2}} * {{NumberPageLink|prev = 0|next = 3|result = Triangular Fibonacci Numbers}} * {{NumberPageLink|prev = 0|next = 3|result = Palindromic Triangular Numbers}} * {{NumberPageLink|prev = 0|next = 3|type = Ramanujan-Nagell Number|cat = Ramanujan-Nagell Numbers}} ==== Next in sequence: $5$ ==== * {{NumberPageLink|prev = 0|next = 5|result = Fibonacci Numbers which equal their Index}} * {{NumberPageLink|prev = 0|next = 5|result = Sequence of Fibonacci Numbers ending in Index}} ==== Next in sequence: $8$ ==== * {{NumberPageLink|prev = 0|next = 8|type = Dudeney Number|cat = Dudeney Numbers}} === Previous in sequence: $2$ === * {{NumberPageLink|prev = 2|next = 3|type = Lucas Number|cat = Lucas Numbers}} === Next in sequence: $2$ === * {{NumberPageLink|next = 2|type = Generalized Pentagonal Number|cat = Generalized Pentagonal Numbers}} * {{NumberPageLink|next = 2|type = Highly Composite Number|cat = Highly Composite Numbers}} * {{NumberPageLink|next = 2|type = Special Highly Composite Number|cat = Special Highly Composite Numbers}} * {{NumberPageLink|next = 2|type = Highly Abundant Number|cat = Highly Abundant Numbers}} * {{NumberPageLink|next = 2|type = Superabundant Number|cat = Superabundant Numbers}} * {{NumberPageLink|next = 2|type = Almost Perfect Number|cat = Almost Perfect Numbers}} * {{NumberPageLink|next = 2|type = Factorial|cat = Factorials}} * {{NumberPageLink|next = 2|type = Superfactorial|cat = Superfactorials}} * {{NumberPageLink|next = 2|type = Fibonacci Number|cat = Fibonacci Numbers}} * {{NumberPageLink|next = 2|type = Catalan Number|cat = Catalan Numbers}} * {{NumberPageLink|next = 2|type = Ulam Number|cat = Ulam Numbers}} * {{NumberPageLink|next = 2|result = Sequence of Powers of 2|cat = Powers of 2}} * {{NumberPageLink|next = 2|result = Powers of 2 with no Zero in Decimal Representation}} * {{NumberPageLink|next = 2|result = Powers of 2 and 5 without Zeroes}} * {{NumberPageLink|next = 2|result = Integer not Expressible as Sum of 5 Non-Zero Squares}} * {{NumberPageLink|next = 2|result = Integers such that all Coprime and Less are Prime}} * {{NumberPageLink|next = 2|type = Göbel's Sequence|cat = Göbel's Sequence}} * {{NumberPageLink|next = 2|type = 3-Göbel Sequence|cat = Göbel's Sequence}} * {{NumberPageLink|next = 2|result = Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes}} * {{NumberPageLink|next = 2|result = Powers of 2 whose Digits are Powers of 2}} * {{NumberPageLink|next = 2|type = Pluperfect Digital Invariant|cat = Pluperfect Digital Invariants}} * {{NumberPageLink|next = 2|result = Factorions Base 10|cat = Factorions}} * {{NumberPageLink|next = 2|result = Consecutive Integers whose Product is Primorial}} * {{NumberPageLink|next = 2|type = Zuckerman Number|cat = Zuckerman Numbers}} * {{NumberPageLink|next = 2|type = Harshad Number|cat = Harshad Numbers}} * {{NumberPageLink|next = 2|result = Square Formed from Sum of 4 Consecutive Binomial Coefficients}} * {{NumberPageLink|next = 2|type = Bell Number|cat = Bell Numbers}} * {{NumberPageLink|next = 2|result = Sequence of Integers whose Cube is Palindromic}} * {{NumberPageLink|next = 2|result = Number of Free Polyominoes}} * {{NumberPageLink|next = 2|type = Mersenne Prime/Historical Note|cat = Mersenne's Assertion}} === Next in sequence: $3$ === * {{NumberPageLink|next = 3|result = Sequence of Powers of 3|cat = Powers of 3}} * {{NumberPageLink|next = 3|type = Lucky Number|cat = Lucky Numbers}} * {{NumberPageLink|next = 3|result = Sequence of Palindromic Lucky Numbers}} * {{NumberPageLink|next = 3|type = Stern Number|cat = Stern Numbers}} * {{NumberPageLink|next = 3|type = Cullen Number|cat = Cullen Numbers}} * {{NumberPageLink|next = 3|result = Numbers whose Sigma is Square|cat = Numbers whose Sigma is Square}} * {{NumberPageLink|next = 3||result = Integers whose Phi times Tau equal Sigma}} * {{NumberPageLink|next = 3|result = Consecutive Integers with Same Euler Phi Value}} * {{NumberPageLink|next = 3|type = Fermat Set}} * {{NumberPageLink|next = 3|result = Numbers such that Tau divides Phi divides Sigma}} * {{NumberPageLink|next = 3|result = Triangular Lucas Numbers}} * {{NumberPageLink|next = 3|result = Representation of 1 as Sum of n Unit Fractions}} * {{NumberPageLink|next = 3|result = Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares}} === Next in sequence: $4$ === * {{NumberPageLink|next = 4|result = Sequence of Powers of 4|cat = Powers of 4}} * {{NumberPageLink|next = 4|type = Tetrahedral Number|cat = Tetrahedral Numbers}} * {{NumberPageLink|next = 4|result = Square and Tetrahedral Numbers}} * {{NumberPageLink|next = 4|type = Trimorphic Number|cat = Trimorphic Numbers}} * {{NumberPageLink|next = 4|type = Powerful Number|cat = Powerful Numbers}} * {{NumberPageLink|next = 4|result = Powers of 2 which are Sum of Distinct Powers of 3}} * {{NumberPageLink|next = 4|result = Squares with No More than 2 Distinct Digits}} === Next in sequence: $5$ === * {{NumberPageLink|next = 5|result = Sequence of Powers of 5|cat = Powers of 5}} * {{NumberPageLink|next = 5|type = Pentagonal Number|cat = Pentagonal Numbers}} * {{NumberPageLink|next = 5|result = Sequence of Palindromic Pentagonal Numbers}} * {{NumberPageLink|next = 5|type = Square Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 5|type = Pentatope Number|cat = Pentatope Numbers}} * {{NumberPageLink|next = 5|type = Automorphic Number|cat = Automorphic Numbers}} * {{NumberPageLink|next = 5|result = Magic Constant of Magic Square}} * {{NumberPageLink|next = 5|result = Odd Numbers not Sum of Prime and Power}} === Next in sequence: $6$ === * {{NumberPageLink|next = 6|result = Sequence of Powers of 6|cat = Powers of 6}} * {{NumberPageLink|next = 6|type = Ore Number|cat = Ore Numbers}} * {{NumberPageLink|next = 6|result = Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors}} * {{NumberPageLink|next = 6|type = Hexagonal Number|cat = Hexagonal Numbers}} * {{NumberPageLink|next = 6|type = Pentagonal Pyramidal Number|cat = Pyramidal Numbers}} === Next in sequence: $7$ === * {{NumberPageLink|next = 7|cat = Powers of 7|result = Sequence of Powers of 7}} * {{NumberPageLink|next = 7|type = Centered Hexagonal Number|cat = Centered Hexagonal Numbers}} * {{NumberPageLink|next = 7|type = Hexagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 7|type = Heptagonal Number|cat = Heptagonal Numbers}} * {{NumberPageLink|next = 7|type = Woodall Number|cat = Woodall Numbers}} * {{NumberPageLink|next = 7|type = Happy Number|cat = Happy Numbers}} * {{NumberPageLink|next = 7|result = Integers whose Sigma Value is Cube}} === Next in sequence: $8$ === * {{NumberPageLink|next = 8|result = Sequence of Powers of 8|cat = Powers of 8}} * {{NumberPageLink|next = 8|type = Cube Number|cat = Cube Numbers}} * {{NumberPageLink|next = 8|result = Cubic Fibonacci Numbers}} * {{NumberPageLink|next = 8|type = Octagonal Number|cat = Octagonal Numbers}} * {{NumberPageLink|next = 8|type = Heptagonal Pyramidal Number|cat = Pyramidal Numbers}} * {{NumberPageLink|next = 8|type = Kaprekar Triple|cat = Kaprekar Numbers}} * {{NumberPageLink|next = 8|result = Sequence of Palindromic Cubes}} === Next in sequence: $9$ === * {{NumberPageLink|next = 9|result = Sequence of Powers of 9|cat = Powers of 9}} * {{NumberPageLink|next = 9|type = Kaprekar Number|cat = Kaprekar Numbers}} * {{NumberPageLink|next = 9|result = Square Numbers whose Sigma is Square|cat = Square Numbers whose Sigma is Square}} * {{NumberPageLink|next = 9|result = Magic Constant of Magic Cube}} === Next in sequence: $10$ === * {{NumberPageLink|next = 10|result = Sum of Terms of Magic Square}} * {{NumberPageLink|next = 10|cat = Powers of 10|result = Sequence of Powers of 10}} * {{NumberPageLink|next = 10|result = Tetrahedral and Triangular Numbers}} * {{NumberPageLink|next = 10|result = Numbers not Sum of Square and Prime}} === Next in sequence: $11$ === * {{NumberPageLink|next = 11|result = Sequence of Powers of 11|cat = Powers of 11}} * {{NumberPageLink|next = 11|type = Repunit|cat = Repunits}} === Next in sequence: $12$ and above === * {{NumberPageLink|next = 12|result = Sequence of Powers of 12|cat = Powers of 12}} * {{NumberPageLink|next = 13|result = Sequence of Powers of 13|cat = Powers of 13}} * {{NumberPageLink|next = 14|result = Sequence of Powers of 14|cat = Powers of 14}} * {{NumberPageLink|next = 15|result = Sequence of Powers of 15|cat = Powers of 15}} * {{NumberPageLink|next = 16|type = Fourth Power|cat = Fourth Powers}} * {{NumberPageLink|next = 16|result = Sequence of Powers of 16|cat = Powers of 16}} * {{NumberPageLink|next = 22|result = Numbers Equal to Number of Digits in Factorial}} * {{NumberPageLink|next = 36|result = Integer both Square and Triangular}} * {{NumberPageLink|next = 36|result = Sum of Terms of Magic Cube}} * {{NumberPageLink|next = 47|result = Numbers whose Cube equals Sum of Sequence of that many Squares|cat = Numbers whose Cube equals Sum of Sequence of that many Squares}} * {{NumberPageLink|next = 55|result = Square Pyramidal and Triangular Numbers}} * {{NumberPageLink|next = 55|type = Square Pyramorphic Number|cat = Square Pyramorphic Numbers}} * {{NumberPageLink|next = 87|result = Integers for which Sigma of Phi equals Sigma}} * {{NumberPageLink|next = 121|type = Wonderful Demlo Number|cat = Wonderful Demlo Numbers}} * {{NumberPageLink|next = 127|type = Obstinate Number|cat = Obstinate Numbers}} * {{NumberPageLink|next = 141|type = Cullen Prime|cat = Cullen Primes}} * {{NumberPageLink|next = 169|result = Sequence of Square Centered Hexagonal Numbers}} * {{NumberPageLink|next = 210|result = Triangular Numbers which are also Pentagonal}} == Historical Note == {{:1/Historical Note}} == Linguistic Note == {{:1/Linguistic Note}} == Sources == * {{BookReference|Les Nombres Remarquables|1983|François Le Lionnais|author2 = Jean Brette|prev = Integral from 0 to 1 of Complete Elliptic Integral of First Kind|next = Trivial Group is Smallest Group}}: $1$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Unit Vector|next = Definition:Universal Quantifier|entry = unity}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Unit Vector|next = Universal Instantiation|entry = unity}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Unit Vector|next = Definition:Universal Gravitational Constant|entry = unity}} Category:Specific Numbers Category:1 f3jrybgjko4vrtj7j704f8zzrwv4hbg


Total Number of Set Partitions/Examples/2/Illustration 0 71136 366804 2018-09-22T08:51:38Z Prime.mover 59 Created page with "== Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $2$ be exemplified by $S = \set {a..." wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $2$ be exemplified by $S = \set {a, b}$. Then the partitions of $S$ are: :$\set {a, b}$ :$\set {a \mid b}$ </onlyinclude> Category:Examples of Set Partitions m79iolwjmaoknsukrz8pneqzcwq0imf


Total Number of Set Partitions/Examples/3/Illustration 0 71137 366808 2018-09-22T08:54:10Z Prime.mover 59 Created page with "== Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $3$ be exemplified by $S = \set {a..." wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $3$ be exemplified by $S = \set {a, b, c}$. Then the partitions of $S$ are: :$\set {a, b, c}$ :$\set {a, b \mid c}$ :$\set {a, c \mid b}$ :$\set {b, c \mid a}$ :$\set {a \mid b \mid c}$ </onlyinclude> Category:Examples of Set Partitions awrrsx8y7vinkak3nia95du5zdfolgp


Total Number of Set Partitions/Examples/4/Illustration 0 71138 418216 418215 2019-08-15T06:28:33Z Prime.mover 59 wikitext text/x-wiki == Example of Total Number of Set Partitions == <onlyinclude> Let a set $S$ of cardinality $4$ be exemplified by $S = \set {a, b, c, d}$. Then the partitions of $S$ are: :$\set {a, b, c, d}$ :$\set {\set a, \set {b, c, d} }$ :$\set {\set b, \set {a, c, d} }$ :$\set {\set c, \set {a, b, d} }$ :$\set {\set d, \set {a, b, c} }$ :$\set {\set {a, b}, \set {c, d} }$ :$\set {\set {a, c}, \set {b, d} }$ :$\set {\set {a, d}, \set {b, c} }$ :$\set {\set a, \set b, \set {c, d} }$ :$\set {\set a, \set c, \set {b, d} }$ :$\set {\set a, \set d, \set {b, c} }$ :$\set {\set b, \set c, \set {a, d} }$ :$\set {\set b, \set d, \set {a, c} }$ :$\set {\set c, \set d, \set {a, b} }$ :$\set {\set a, \set b, \set c, \set d}$ </onlyinclude> == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Definition:Set Partition/Definition 1|next = Set Partition/Examples/Partition into Singletons}}: $\S 6$. Indexed families; partitions; equivalence relations: Example $6.2$ Category:Examples of Set Partitions g7vdhma5nj19r1y3o9j4rnx5w4mq25j


Modulo Addition/Cayley Table/Modulo 6

Modulo Addition/Cayley Table/Modulo 6 0 71211 379077 378169 2018-11-29T13:28:19Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Modulo Addition == The additive group of integers modulo $m$ can be described by showing its Cayley table. This one is for modulo $6$: <onlyinclude> :$\begin{array}{r|rrrrrr} \struct {\Z_6, +_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 1 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 \\ \eqclass 2 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 \\ \eqclass 3 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 \\ \eqclass 4 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 \\ \eqclass 5 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 \\ \end{array}$ </onlyinclude> which can also be presented: :$\begin{array}{r|rrrrrr} +_6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5  \\ 1 & 1 & 2 & 3 & 4 & 5 & 0 \\ 2 & 2 & 3 & 4 & 5 & 0 & 1 \\ 3 & 3 & 4 & 5 & 0 & 1 & 2 \\ 4 & 4 & 5 & 0 & 1 & 2 & 3 \\ 5 & 5 & 0 & 1 & 2 & 3 & 4 \\ \end{array}$ == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Symmetric Group on 3 Letters/Cayley Table|next = Alternating Group on 4 Letters/Cayley Table}}: Tables: $2$. Cyclic group of order $6$ * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Definition:Multiplication/Modulo Multiplication/Definition 3|next = Modulo Multiplication/Cayley Table/Modulo 6}}: $\S 2$: Example $2.3$ Category:Examples of Cayley Tables Category:Groups of Order 6 Category:Examples of Additive Groups of Integers Modulo m hjmtao7z2dr3n1jhme9h0i1x6pjfg32


Modulo Multiplication/Cayley Table/Modulo 6 0 71213 397332 370857 2019-03-25T21:58:59Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Modulo Multiplication == The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table. This one is for modulo $6$: <onlyinclude> :$\begin{array} {r|rrrrrr} \struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6  \\ \eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 \\ \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 \\ \eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 \\ \eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6 \\ \end{array}$ which can also be presented: :$\begin{array} {r|rrrrrr} \times_6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0  \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \\ \end{array}$ </onlyinclude> == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Modulo Addition/Cayley Table/Modulo 6|next = Modulo Multiplication is Associative/Proof 2}}: $\S 2$: Example $2.3$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Equivalence Relation on Square Matrices induced by Positive Integer Powers|next = Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6}}: Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $7$ Category:Examples of Cayley Tables Category:Modulo Multiplication qdh7ih2623tp3ft6811sln6l30hf97q


Cauchy Sequences form Ring with Unity/Corollary 0 71225 472816 472815 2020-06-09T06:45:06Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cauchy Sequences form Ring with Unity == <onlyinclude> Let $\struct {F, +, \circ, \norm {\, \cdot \,} }$ be a valued field. Let $\struct {F^\N, +, \circ}$ be the commutative ring of sequences over $F$ with unity $\tuple {1, 1, 1, \dotsc}$. {{explain|Has it been proved that $\struct {F^\N, +, \circ}$ is actually a commutative ring?}} Let $\CC \subset F^\N$ be the set of Cauchy sequences on $F$. Then: :$\struct {\CC, +, \circ}$ is a commutative subring of $F^\N$ with unity $\tuple {1, 1, 1, \dotsc}$. </onlyinclude> == Proof == By Cauchy Sequences form Ring with Unity, $\struct {\CC, +, \circ}$ is a subring of $F^\N$. We have that $\circ$ is commutative on $F^\N$. Hence by Restriction of Commutative Operation is Commutative the restriction of $\circ$ to $\CC$ is commutative. == Sources == * {{BookReference|p-adic Numbers: An Introduction|1997|Fernando Q. Gouvea}}: $\S 3.2$: Completions Category:Cauchy Sequences in Normed Division Rings shjh29azsntz6pkub11ghpuvur8rmlw


Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97 0 71245 367522 367490 2018-09-27T15:38:12Z Prime.mover 59 wikitext text/x-wiki == Example of Modulo Arithmetic == <onlyinclude> The inverse of $41$ under multiplication modulo $97$ is given by: :${\eqclass {41} {97} }^{-1} = 71$ </onlyinclude> === Solution to $41 x \equiv 2 \pmod {97}$ === {{:Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97}} == Proof == From Ring of Integers Modulo Prime is Field, multiplication modulo $97$ has an inverse for all $x \in \Z_{97}$ where $x \ne 0$. Using Euclid's Algorithm: {{begin-eqn}} {{eqn | n = 1       | l = 97       | r = 2 \times 41 + 15 }} {{eqn | n = 2       | l = 41       | r = 2 \times 15 + 11 }} {{eqn | n = 3       | l = 15       | r = 11 + 4 }} {{eqn | n = 4       | l = 11       | r = 2 \times 4 + 3 }} {{eqn | n = 5       | l = 4       | r = 3 + 1 }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = 1       | r = 4 - 3       | c = from $(5)$ }} {{eqn | r = 4 - \paren {11 - 2 \times 4}       | c = from $(4)$ }} {{eqn | r = 3 \times 4 - 11       | c =  }} {{eqn | r = 3 \times \paren {15 - 11} - 11       | c = from $(3)$ }} {{eqn | r = 3 \times 15 - 4 \times 11       | c =  }} {{eqn | r = 3 \times 15 - 4 \times \paren {41 - 2 \times 15}       | c = from $(2)$ }} {{eqn | r = 11 \times 15 - 4 \times 41       | c =  }} {{eqn | r = 11 \times \paren {97 - 2 \times 14} - 4 \times 41       | c = from $(1)$ }} {{eqn | r = 11 \times 97 - 26 \times 41       | c =  }} {{end-eqn}} So: : $\paren {-26} \times 41 \equiv 1 \pmod {97}$ {{begin-eqn}} {{eqn | l = \paren {-26} \times 41       | o = \equiv       | r = 1       | rr= \pmod {97}       | c =  }} {{eqn | l = 71 \times 41       | o = \equiv       | r = 1       | rr= \pmod {97}       | c = as $-26 \equiv 71 \pmod {97}$ because $26 + 71 = 97$ }} {{eqn | ll= \leadsto       | l = {\eqclass {41} {97} }^{-1}       | r = \eqclass {71} {97}       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6|next = Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97}}: Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $8$ Category:Examples of Modulo Arithmetic Category:Multiplicative Inverse of 41 Modulo 97 aj7quf3yc5k62f9zgwb5kwrh5za2ik7


Real Square Function is not Injective

Real Square Function is not Injective 0 71342 376794 376744 2018-11-15T16:13:56Z Prime.mover 59 wikitext text/x-wiki == Example of Mapping which is Not an Injection == <onlyinclude> Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Then $f$ is not an injection. </onlyinclude> == Proof == For $f$ to be an injection, it would be necessary that: :$\forall x_1, x_2 \in \R: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$ By definition of the squaring operation, we have: :$\map f x = \map f {-x}$ But unless $x = 0$ it is not the case that $x = -x$. Hence $f$ is not an injection. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Injection/Definition 1 a|next = Injection/Examples/Cube Function}}: $\S 22$: Injections; bijections; inverse of a bijection Category:Examples of Injections Category:Square Function rnjglt6paz96657wm169cjbl4y054mu


Image of Mapping/Examples/Image of x^2-4x+5 0 71455 427644 368749 2019-09-25T15:48:30Z Prime.mover 59 wikitext text/x-wiki == Example of Image of Element under Mapping == <onlyinclude> Let $f: \R \to \R$ be the mapping defined as: :$\forall x \in \R: \map f x = x^2 - 4 x + 5$ The image of $f$ is the unbounded closed interval: :$\Img f = \hointr 1 \to$ and so $f$ is not a surjection. </onlyinclude> === Graphical Representation of $\map f x = x^2 - 4 x + 5$ === {{:Image of Mapping/Examples/Image of x^2-4x+5/Graph}} == Proof == By differentiating $x^2 - 4 x + 5$ twice {{WRT|Differentiation}} $x$: :$f' = 2 x - 4$ :$f' = 2 x - 4$ {{begin-eqn}} {{eqn | l = f'       | r = 2 x - 4 }} {{eqn | l = f''       | r = 2 }} {{end-eqn}} Equating $f'$ to $0$, a stationary point is found at $x = 2$. Inspecting the sign of $f''$, it is noted that $f'$ is increasing everywhere. Hence the stationary point at $x = 2$ is a minimum of $\Img f$. This is the only stationary point, so it can be stated that '''the''' minimum of $f$ occurs at $x = 2$. We have that: :$f \paren 2 = 2^2 - 4 \times 2 + 5 = 4 - 8 + 5 = 1$ As $f$ is strictly increasing on $x > 2$ and strictly decreasing on $x < 2$, it is seen that $f$ is unbounded above. Thus: :$\Img f = \hointr 1 \to$ {{qed}} == Also see == * Bijective Restrictions of $\map f x = x^2 - 4 x + 5$ == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Cantor's Theorem|next = Bijective Restriction/Examples/x^2-4x+5}}: Chapter $4$: Mappings: Exercise $12 \ \text{(ii)}$ Category:Examples of Images under Mappings 4c0225x2ed0rbwxsccxhz3tccv0cdge


Subsemigroup/Examples/2x2 Matrices with One Non-Zero Entry 0 71509 369174 369160 2018-10-05T22:25:59Z Prime.mover 59 wikitext text/x-wiki == Example of Subsemigroup == <onlyinclude> Let $\struct {S, \times}$ be the semigroup formed by the set of order $2$ square matrices over the real numbers $R$ under (conventional) matrix multiplication. Let $T$ be the subset of $S$ consisting of the matrices of the form $\begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ for $x \in \R$. Then $\struct {T, \times}$ is a subsemigroup of $\struct {S, \times}$. </onlyinclude> == Proof == From the Subsemigroup Closure Test it is sufficient to demonstrate that $\struct {T, \times}$ is closed. Let $A = \begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} y & 0 \\ 0 & 0 \end{bmatrix}$. Then: {{begin-eqn}} {{eqn | l = A B       | r = \begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} y & 0 \\ 0 & 0 \end{bmatrix}       | c =  }} {{eqn | r = \begin{bmatrix} x y + 0 \times 0 & x \times 0 + 0 \times 0 \\ 0 \times y + 0 \times 0 & 0 \times 0 + 0 \times 0 \end{bmatrix}       | c = {{Defof|Matrix Product (Conventional)}} }} {{eqn | r = \begin{bmatrix} x y & 0 \\ 0 & 0 \end{bmatrix}       | c =  }} {{eqn | 0 = \in       | r = T       | c =  }} {{end-eqn}} Hence the result. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Subsemigroup Closure Test|next = Identity of Submonoid is not necessarily Identity of Monoid}}: $\S 32$ Identity element and inverses Category:Examples of Subsemigroups 18jb5pq679bn9stqcgsimhlezkawejn


Semigroup/Examples/x+y-xy on Integers

Semigroup/Examples/x+y-xy on Integers 0 71513 369176 369175 2018-10-05T22:31:28Z Prime.mover 59 wikitext text/x-wiki == Example of Semigroup == <onlyinclude> Let $\circ: \Z \times \Z$ be the operation defined on the integers $\Z$ as: :$\forall x, y \in \Z: x \circ y := x + y - x y$ Then $\struct {\Z, \circ}$ is a semigroup. </onlyinclude> == Proof == We have that: :$\forall x, y \in \Z: x \circ y \in \Z$ and so $\struct {\Z, \circ}$ is closed. Now let $x, y, z \in \Z$. We have: {{begin-eqn}} {{eqn | l = x \circ \paren {y \circ z}       | r = x + \paren {y \circ z} - x \paren {y \circ z}       | c = Definition of $\circ$ }} {{eqn | r = x + \paren {y + z - y z} - x \paren {y + z - y z}       | c = Definition of $\circ$ }} {{eqn | r = x + y + z - y z - x y - x z + x y z       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \paren {x \circ y} \circ z       | r = \paren {x \circ y} + z - \paren {x \circ y} z       | c = Definition of $\circ$ }} {{eqn | r = \paren {x + y - x y} + z - \paren {x + y - x y} z       | c = Definition of $\circ$ }} {{eqn | r = x + y - x y + z - x z - y z + x y z       | c =  }} {{end-eqn}} As can be seen by inspection: :$x \circ \paren {y \circ z} = \paren {x \circ y} \circ z$ and so $\circ$ is associative. The result follows by definition of semigroup. {{qed}} == Also see == * Inclusion-Exclusion Principle (think about why) == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Identity of Submonoid is not necessarily Identity of Monoid|next = Subsemigroup/Examples/x+y-xy on Integers}}: Chapter $5$: Semigroups: Exercise $1$ Category:Examples of Semigroups 29jgsmiq5wfzvoahsghgaqhysfrstze


Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers 0 71674 370495 370432 2018-10-11T19:59:37Z Prime.mover 59 wikitext text/x-wiki == Examples of Order of Group Element == <onlyinclude> Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$. The order of $2$ in $\struct {\R_{\ne 0}, \times}$ is infinite. </onlyinclude> == Proof == From Real Multiplication Identity is One, the identity of $\struct {\R_{\ne 0}, \times}$ is $1$. There exists no $n \in \Z_{\ge 0}$ such that $2^n = 1$. Hence the result by definition of infinite order element. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Definition:Order of Group Element/Also known as|next = Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers}}: $\S 38$. Period of an element: Illustrations: $\text{(i)}$ Category:Examples of Order of Group Elements Category:Multiplicative Group of Real Numbers r8303q13pbsr1eujhsy2zscbynoqbsf


Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers 0 71675 370340 370337 2018-10-11T06:38:37Z Prime.mover 59 Prime.mover moved page Order of Group Element/Examples/Gaussian Integer in Multiplicative Group of Complex Numbers to Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers wikitext text/x-wiki == Examples of Order of Group Element == <onlyinclude> Consider the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$. The order of $i$ in $\struct {\C_{\ne 0}, \times}$ is $4$. </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = i^1       | r = 1       | c =  }} {{eqn | l = i^2       | r = -1       | c =  }} {{eqn | l = i^3       | r = -i       | c =  }} {{eqn | l = i^4       | r = 1       | c =  }} {{end-eqn}} Hence the result by definition of order of group element. {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Order of Group Element/Examples/Element of Multiplicative Group of Real Numbers|next = Identity is Only Group Element of Order 1}}: $\S 38$. Period of an element: Illustrations: $\text{(ii)}$ Category:Examples of Order of Group Elements oh68f40teq2kjhn3mrnyhe3jlurecwi


Quaternion Group/Cayley Table

Quaternion Group/Cayley Table 0 71742 387867 374905 2019-01-15T13:02:12Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Quaternion Group== The Cayley table for the quaternion group given with the group presentation: :$Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ can be presented as: <onlyinclude> :$\begin{array}{r|rrrrrrrr}       & e     & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\ \hline e     & e     & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\ a     & a     & a^2   & a^3   & e     & a b   & a^2 b & a^3 b & b     \\ a^2   & a^2   & a^3   & e     & a     & a^2 b & a^3 b & b     & a b   \\ a^3   & a^3   & e     & a     & a^2   & a^3 b & b     & a b   & a^2 b \\ b     & b     & a^3 b & a^2 b & a b   & a^2   & a     & e     & a^3   \\ a b   & a b   & b     & a^3 b & a^2 b & a^3   & a^2   & a     & e     \\ a^2 b & a^2 b & a b   & b     & a^3 b & e     & a^3   & a^2   & a     \\ a^3 b & a^3 b & a^2 b & a b   & b     & a     & e     & a^3   & a^2  \end{array}$ </onlyinclude> === Coset Decomposition of $\set {e, a^2}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen a^2$ gives: {{:Quaternion Group/Cayley Table/Coset Decomposition of (e, a^2)}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Quaternion Group|next = Quaternion Group is Hamiltonian}}: Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \iota$ Category:Quaternion Group Category:Examples of Cayley Tables lnm51jeg7pcggtfeaiqu9z7kxcezvi6


Quaternion Group/Complex Matrices/Cayley Table 0 71744 370945 370909 2018-10-13T22:01:19Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Quaternion Group== The Cayley table for the quaternion group: :$Q = \Dic 2 = \set {\mathbf 1, -\mathbf 1, \mathbf i, -\mathbf i, \mathbf j, -\mathbf j, \mathbf k, -\mathbf k}$ under the operation of conventional matrix multiplication, where: :$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ can be presented as: <onlyinclude> :$\begin{array}{r|rrrrrrrr}       &  \mathbf 1 &  \mathbf i & -\mathbf 1 & -\mathbf i &  \mathbf j &  \mathbf k & -\mathbf j & -\mathbf k \\ \hline  \mathbf 1 &  \mathbf 1 &  \mathbf i & -\mathbf 1 & -\mathbf i &  \mathbf j &  \mathbf k & -\mathbf j & -\mathbf k \\  \mathbf i &  \mathbf i & -\mathbf 1 & -\mathbf i &  \mathbf 1 &  \mathbf k & -\mathbf j & -\mathbf k &  \mathbf j \\ -\mathbf 1 & -\mathbf 1 & -\mathbf i &  \mathbf 1 &  \mathbf i & -\mathbf j & -\mathbf k &  \mathbf j &  \mathbf k \\ -\mathbf i & -\mathbf i &  \mathbf 1 &  \mathbf i & -\mathbf 1 & -\mathbf k &  \mathbf j &  \mathbf k & -\mathbf j \\  \mathbf j &  \mathbf j & -\mathbf k & -\mathbf j &  \mathbf k & -\mathbf 1 &  \mathbf i &  \mathbf 1 & -\mathbf i \\  \mathbf k &  \mathbf k &  \mathbf j & -\mathbf k & -\mathbf j & -\mathbf i & -\mathbf 1 &  \mathbf i &  \mathbf 1 \\ -\mathbf j & -\mathbf j &  \mathbf k &  \mathbf j & -\mathbf k &  \mathbf 1 & -\mathbf i & -\mathbf 1 &  \mathbf i \\ -\mathbf k & -\mathbf k & -\mathbf j &  \mathbf k &  \mathbf j &  \mathbf i &  \mathbf 1 & -\mathbf i & -\mathbf 1 \end{array}$ </onlyinclude> == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Order of Power of Group Element/Examples/Powers of Element of Order 20|next = Order of Elements in Quaternion Group}}: Chapter $6$: An Introduction to Groups: Exercise $14$ Category:Quaternion Group Category:Examples of Cayley Tables a8334fcqkltaejdxgcf2c1z0aauwyi2


Order of Subgroup Product/Lemma

Order of Subgroup Product/Lemma 0 71989 372771 2018-10-23T14:20:28Z Prime.mover 59 Created page with "== Lemma for Order of Subgroup Product == <onlyinclude> Let $h_1, h_2 \in H$.  Then: :$h_1 K = h_2 K$ {{iff}}: :$h_1$ and $h_2$ are in the same Definition:Left Coset|lef..." wikitext text/x-wiki == Lemma for Order of Subgroup Product == <onlyinclude> Let $h_1, h_2 \in H$. Then: :$h_1 K = h_2 K$ {{iff}}: :$h_1$ and $h_2$ are in the same left coset of $H \cap K$. </onlyinclude> == Proof == Let $h_1, h_2 \in H$. Then: {{begin-eqn}} {{eqn | l = h_1 K       | r = h_2 K       | c =  }} {{eqn | ll= \iff       | l = h_1^{-1} h_2       | o = \in       | r = K       | c = Left Cosets are Equal iff Product with Inverse in Subgroup }} {{eqn | ll= \iff       | l = h_1^{-1} h_2       | o = \in       | r = H \cap K       | c = {{Defof|Set Intersection}} }} {{eqn | ll= \iff       | l = h_1 \paren {H \cap K}       | r = h_2 \paren {H \cap K}       | c = Left Cosets are Equal iff Product with Inverse in Subgroup }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Intersection of Left Cosets of Subgroups is Left Coset of Intersection|next = Order of Subgroup Product/Proof 2}}: Chapter $7$: Cosets and Lagrange's Theorem: Exercise $10$ Category:Order of Subgroup Product 8ppal07pfp8shxynzdiwus7k1ph9xcu


Symmetry Group of Regular Hexagon/Group Action on Vertices 0 72174 444487 374648 2020-01-22T13:40:32Z Prime.mover 59 wikitext text/x-wiki == Group Action of Symmetry Group of Regular Hexagon == Let $\HH = ABCDEF$ be a regular hexagon. Let $D_6$ denote the symmetry group of $\HH$. :520px Let $e$ denote the identity mapping Let $\alpha$ denote rotation of $\HH$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$) Let $\beta$ denote reflection of $\HH$ in the $AD$ axis. <onlyinclude> $D_6$ acts on the vertices of $\HH$ according to this table: :$\begin{array}{cccccccccccc}  e & \alpha & \alpha^2 & \alpha^3 & \alpha^4 & \alpha^5 & \beta & \alpha \beta & \alpha^2 \beta & \alpha^3 \beta & \alpha^4 \beta & \alpha^5 \beta \\ \hline A & B & C & D & E & F & A & B & C & D & E & F \\ B & C & D & E & F & A & F & A & B & C & D & E \\ C & D & E & F & A & B & E & F & A & B & C & D \\ D & E & F & A & B & C & D & E & F & A & B & C \\ E & F & A & B & C & D & C & D & E & F & A & B \\ F & A & B & C & D & E & B & C & D & E & F & A \\ \end{array}$ </onlyinclude> == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Symmetry Group of Regular Hexagon|next = Symmetry Group of Regular Hexagon/Examples/Subgroup that Fixes C}}: Chapter $2$: Subgroups and Cosets: $\S 35 \zeta$ Category:Symmetry Group of Regular Hexagon m0iwr36mc45sx35m6unnqrg85efbqc2


Composition of Distance-Preserving Mappings is Distance-Preserving 0 72279 375364 2018-11-08T10:30:08Z Leigh.Samphier 3031 Created page with "{{Proofread}} == Theorem == <onlyinclude> Let: : $\struct{X_1, d_1}$ : $\struct{X_2, d_2}$ : $\struct{X_3, d_3}$ be metric spaces.  Let: : $\phi: \..." wikitext text/x-wiki {{Proofread}} == Theorem == <onlyinclude> Let: : $\struct{X_1, d_1}$ : $\struct{X_2, d_2}$ : $\struct{X_3, d_3}$ be metric spaces. Let: : $\phi: \struct{X_1, d_1} \to \struct{X_2, d_2}$ : $\psi: \struct{X_2, d_2} \to \struct{X_3, d_3}$ be distance-preserving mappings. Then the composite of $\phi$ and $\psi$ is also a distance-preserving mapping. </onlyinclude> == Proof == Let $x,y \in X_1$ then: {{begin-eqn}} {{eqn|l= d_1 \paren {x,y}      |r= d_2 \paren {\map \phi x, \map \phi y }      |c= $\phi$ is a distance-preserving mapping }} {{eqn|r= d_3 \paren {\map \psi {\map \phi x}, \map \psi {\map \phi y} }      |c= $\psi$ is a distance-preserving mapping }} {{eqn|r= d_3 \paren {\map {\psi \circ \phi} x, \map {\psi \circ \phi} y }      |c= Definition of composite mappings }} {{end-eqn}} By the definition of a distance-preserving mapping then $\psi \circ \phi$ is distance-preserving. {{qed}} Category:Metric Spaces hfxs3l40pav13ef7tjwiauyc64ky2dk


First Sylow Theorem/Corollary

First Sylow Theorem/Corollary 0 72298 375516 375507 2018-11-09T08:07:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to First Sylow Theorem == <onlyinclude> Let $p$ be a prime number. Let $G$ be a group. Let: :$p^n \divides \order G$ where: :$\order G$ denotes the order of $G$ :$n$ is a positive integer. Then $G$ has at least one subgroup of order $p$. </onlyinclude> == Proof 1 == {{:First Sylow Theorem/Corollary/Proof 1}} == Proof 2 == This result can also be proved directly: {{:First Sylow Theorem/Corollary/Proof 2}} == Sources == * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 1|next = Sylow Theorems/Historical Note}}: $\S 44$. Some consequences of Lagrange's Theorem Category:Sylow Theorems Category:First Sylow Theorem ney8asccuu12b8eb2wiwnl2i3np323q


Real Square Function is not Surjective

Real Square Function is not Surjective 0 72448 376766 376755 2018-11-15T12:35:50Z Prime.mover 59 wikitext text/x-wiki == Example of Mapping which is Not a Surjection == <onlyinclude> Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Then $f$ is not a surjection. </onlyinclude> == Proof == For $f$ to be a surjection, it would be necessary that: :$\forall y \in \R: \exists x \in \R: \map f x = y$ However from Square of Real Number is Non-Negative: :$\forall y \in \R_{< 0}: \nexists x \in \R: \map f x = y$ Hence $f$ is not a surjection. {{qed}} Category:Examples of Surjections Category:Square Function jirhg29kr6a3z2p55pv7z9u7rl9en6t


Monoid/Examples/x+y+xy on Reals

Monoid/Examples/x+y+xy on Reals 0 72644 377613 377611 2018-11-20T22:53:40Z Prime.mover 59 wikitext text/x-wiki == Example of Monoid == <onlyinclude> Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as: :$\forall x, y \in \R: x \circ y := x + y + x y$ Then $\struct {\R, \circ}$ is a monoid whose identity is $0$. </onlyinclude> == Proof == We have that: :$\forall x, y \in \R: x \circ y \in \R$ and so $\struct {\R, \circ}$ is closed. Now let $x, y, z \in \R$. We have: {{begin-eqn}} {{eqn | l = x \circ \paren {y \circ z}       | r = x + \paren {y \circ z} + x \paren {y \circ z}       | c = Definition of $\circ$ }} {{eqn | r = x + \paren {y + z + y z} + x \paren {y + z + y z}       | c = Definition of $\circ$ }} {{eqn | r = x + y + z + y z + x y + x z + x y z       | c =  }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \paren {x \circ y} \circ z       | r = \paren {x \circ y} + z + \paren {x \circ y} z       | c = Definition of $\circ$ }} {{eqn | r = \paren {x + y + x y} + z + \paren {x + y + x y} z       | c = Definition of $\circ$ }} {{eqn | r = x + y + x y + z + x z + y z + x y z       | c =  }} {{end-eqn}} As can be seen by inspection: :$x \circ \paren {y \circ z} = \paren {x \circ y} \circ z$ and so $\circ$ is associative. Then we have: {{begin-eqn}} {{eqn | l = x \circ 0       | r = x + 0 + x \times 0       | c = Definition of $\circ$ }} {{eqn | r = x       | c =  }} {{eqn | r = 0 + x + 0 \times x       | c =  }} {{eqn | r = 0 \circ x       | c = Definition of $\circ$ }} {{end-eqn}} The result follows by definition of monoid. {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Product of Semigroup Element with Right Inverse is Idempotent|next = Group/Examples/x+y+xy over Reals less -1}}: Chapter $4$. Groups: Exercise $11$ Category:Examples of Monoids 8k8qism2j4hs6tzrbri531k0xz0ye4e


Symmetric Group on 3 Letters/Subgroups

Symmetric Group on 3 Letters/Subgroups 0 72657 387850 386199 2019-01-15T10:15:05Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Symmetric Group on $3$ Letters == Let $S_3$ denote the Symmetric Group on $3$ Letters, whose Cayley table is given as: {{:Symmetric Group on 3 Letters/Cayley Table}} <onlyinclude> The subsets of $S_3$ which form subgroups of $S_3$ are: {{begin-eqn}} {{eqn | o =        | r = S_3 }} {{eqn | o =        | r = \set e }} {{eqn | o =        | r = \set {e, \tuple {123}, \tuple {132} } }} {{eqn | o =        | r = \set {e, \tuple {12} } }} {{eqn | o =        | r = \set {e, \tuple {13} } }} {{eqn | o =        | r = \set {e, \tuple {23} } }} {{end-eqn}} </onlyinclude> == Examples == {{:Symmetric Group on 3 Letters/Subgroups/Examples}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Integer Multiples under Addition form Subgroup of Integers|next = Intersection of Subgroups/General Result}}: $\S 5.2$. Subgroups: Example $93$ * {{BookReference|Problems in Group Theory|1967|John D. Dixon|prev = Symmetric Group on 3 Letters/Group Presentation|next = Quaternion Group/Complex Matrices}}: $1$: Subgroups: Problem $1.1$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Conjugate of Subgroup is Subgroup|next = Symmetric Group on 3 Letters/Normal Subgroups}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $2$ Category:Symmetric Group on 3 Letters ox31arglp9woaqb3klh0cjskly338vd


Alternating Group on 4 Letters

Alternating Group on 4 Letters 0 72718 378592 378497 2018-11-26T21:28:05Z Prime.mover 59 wikitext text/x-wiki == Group Example == Let $S_4$ denote the symmetric group on $4$ letters. <onlyinclude> The '''alternating group on $4$ letters''' $A_4$ is the kernel of the mapping $\sgn: S_4 \to C_2$. </onlyinclude> === Cycle Notation === It can be expressed in the form of permutations given in cycle notation as follows: {{:Alternating Group on 4 Letters/Cycle Notation}} === Cayley Table === The Cayley table of $A_4$ can be written: {{:Alternating Group on 4 Letters/Cayley Table}} == Order of Elements == {{:Alternating Group on 4 Letters/Order of Elements}} == Subgroups == {{:Alternating Group on 4 Letters/Subgroups}} == Normality of Subgroups == {{:Alternating Group on 4 Letters/Normality of Subgroups}} == Conjugacy Classes == {{:Alternating Group on 4 Letters/Conjugacy Classes}} Category:Groups of Order 12 Category:Alternating Group on 4 Letters Category:Examples of Alternating Groups b8c53avi3mg9mezem94330pcr7iayd8


Alternating Group on 4 Letters/Cayley Table 0 72721 379079 378599 2018-11-29T13:33:18Z Prime.mover 59 wikitext text/x-wiki == Cayley Table of Alternating Group on $4$ Letters == The Cayley table of the alternating group on $4$ letters can be written: <onlyinclude> :$\begin{array}{c|cccc|cccc|cccc} \circ & e & t & u & v & a & b & c & d & p & q & r & s \\ \hline e & e & t & u & v & a & b & c & d & p & q & r & s \\ t & t & e & v & u & b & a & d & c & q & p & s & r \\ u & u & v & e & t & c & d & a & b & r & s & p & q \\ v & v & u & t & e & d & c & b & a & s & r & q & p \\ \hline a & a & c & d & b & p & r & s & q & e & u & v & t \\ b & b & d & c & a & q & s & r & p & t & v & u & e \\ c & c & a & b & d & r & p & q & s & u & e & t & v \\ d & d & b & a & c & s & q & p & r & v & t & e & u \\ \hline p & p & s & q & r & e & v & t & u & a & d & b & c \\ q & q & r & p & s & t & u & e & v & b & c & a & d \\ r & r & q & s & p & u & t & v & e & c & b & d & a \\ s & s & p & r & q & v & e & u & t & d & a & c & b \\ \end{array}$ </onlyinclude> where the expression for $A_4$ in cycle notation is given as: {{:Alternating Group on 4 Letters/Cycle Notation}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Modulo Addition/Cayley Table/Modulo 6|next = Definition:Klein Four-Group}}: Tables: $3$. Alternating group $\map A 4$ Category:Examples of Cayley Tables Category:Alternating Group on 4 Letters nwy2h2z2dzkaiehq36hkylmxreidu1z


Alternating Group on 4 Letters/Cycle Notation 0 72722 388451 388450 2019-01-19T06:15:16Z Prime.mover 59 wikitext text/x-wiki == Cycle Notation for Alternating Group on $4$ Letters == The alternating group on $4$ letters can be given in cycle notation as follows: <onlyinclude> {{begin-eqn}} {{eqn | l = e       | o = :=       | r = \text { the identity mapping} }} {{eqn | l = t       | o = :=       | r = \tuple {1 2} \tuple {3 4} }} {{eqn | l = u       | o = :=       | r = \tuple {1 3} \tuple {2 4} }} {{eqn | l = v       | o = :=       | r = \tuple {1 4} \tuple {2 3} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = a       | o = :=       | r = \tuple {1 2 3} }} {{eqn | l = b       | o = :=       | r = \tuple {1 3 4} }} {{eqn | l = c       | o = :=       | r = \tuple {2 4 3} }} {{eqn | l = d       | o = :=       | r = \tuple {1 4 2} }} {{end-eqn}} {{begin-eqn}} {{eqn | l = p       | o = :=       | r = \tuple {1 3 2} }} {{eqn | l = q       | o = :=       | r = \tuple {2 3 4} }} {{eqn | l = r       | o = :=       | r = \tuple {1 2 4} }} {{eqn | l = s       | o = :=       | r = \tuple {1 4 3} }} {{end-eqn}} </onlyinclude> == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 4 Letters/Subgroups/Examples/Disjoint Transpositions|next = Alternating Group on 4 Letters/Order of Elements}}: Chapter $9$: Permutations: Exercise $3$ Category:Alternating Group on 4 Letters 00uol1mkvs05aj9c2bxpvntjgd8kzuy


Symmetric Group on 3 Letters/Normal Subgroups 0 72767 387861 387857 2019-01-15T11:18:01Z Prime.mover 59 wikitext text/x-wiki == Normal Subgroups of the Symmetric Group on 3 Letters == Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as: {{:Symmetric Group on 3 Letters/Cayley Table}} <onlyinclude> Consider the subgroups of $S_3$: {{:Symmetric Group on 3 Letters/Subgroups}} Of those, the normal subgroups in $S_3$ are: :$S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$ </onlyinclude> == Proof == $S_3$ itself is normal in $S_3$ by Group is Normal in Itself. $\set e$ is normal in $S_3$ by Trivial Subgroup is Normal. $\set {e, \tuple {12} }$: {{begin-eqn}} {{eqn | l = \tuple {123} \tuple {13} \tuple {123}^{-1}       | r = \tuple {123} \tuple {13} \tuple {132}       | c =  }} {{eqn | r = \tuple {123} \tuple {12}       | c =  }} {{eqn | r = \tuple {23}       | c =  }} {{eqn | o = \notin       | r = \set {e, \tuple {12} }       | c =  }} {{end-eqn}} Hence $\set {e, \tuple {12} }$ is not normal in $S_3$. $\set {e, \tuple {23} }$: {{begin-eqn}} {{eqn | l = \tuple {123} \tuple {23} \tuple {123}^{-1}       | r = \tuple {123} \tuple {23} \tuple {132}       | c =  }} {{eqn | r = \tuple {123} \tuple {13}       | c =  }} {{eqn | r = \tuple {12}       | c =  }} {{eqn | o = \notin       | r = \set {e, \tuple {23} }       | c =  }} {{end-eqn}} Hence $\set {e, \tuple {23} }$ is not normal in $S_3$. $\set {e, \tuple {13} }$: {{begin-eqn}} {{eqn | l = \tuple {123} \tuple {13} \tuple {123}^{-1}       | r = \tuple {123} \tuple {13} \tuple {132}       | c =  }} {{eqn | r = \tuple {123} \tuple {12}       | c =  }} {{eqn | r = \tuple {23}       | c =  }} {{eqn | o = \notin       | r = \set {e, \tuple {13} }       | c =  }} {{end-eqn}} Hence $\set {e, \tuple {13} }$ is not normal in $S_3$. $\set {e, \tuple {123}, \tuple {132} }$: We have that $\set {e, \tuple {123}, \tuple {132} }$ is the set of even permutations of $S_3$. Any permutation of the form $\alpha \pi \alpha^{-1}$, for $\pi$ even, is also even. Thus: :$\forall \alpha \in S_3: \alpha \pi \alpha^{-1} \in \set {e, \tuple {123}, \tuple {132} }$ Hence $\set {e, \tuple {123}, \tuple {132} }$ is normal in $S_3$. {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Subgroup is Superset of Conjugate iff Normal|next = Group is Normal in Itself}}: $\S 6.6$. Normal subgroups: Example $122$ * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 3 Letters/Subgroups|next = Subgroups of Quaternion Group}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $2$ Category:Symmetric Group on 3 Letters Category:Examples of Normal Subgroups gr8pg2feru0djipvuiyeu3gr5xtagfi


Modulo Addition/Cayley Table/Modulo 3

Modulo Addition/Cayley Table/Modulo 3 0 72769 463911 463905 2020-04-24T06:18:00Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Addition Modulo $3$ == The additive group of integers modulo $3$ can be described by showing its Cayley table: <onlyinclude> :$\begin{array}{r|rrr} \struct {\Z_3, +_3} & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 0 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$ </onlyinclude> It can also be presented: :$\begin{array}{r|rrr} +_3 & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \\ \end{array}$ == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Quotient Group of Integers by Multiples|next = Klein Four-Group is Normal in A4}}: $\S 6.7$. Quotient groups: Example $126$ * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Parity of Integer|next = Definition:Quotient Group/Motivation}}: Chapter $\text{II}$: Groups: A Little Number Theory * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Integers Modulo m under Addition form Abelian Group|next = Mathematician:Arthur Cayley}}: Chapter $2$: Maps and relations on sets: Example $2.33$ Category:Examples of Cayley Tables Category:Examples of Additive Groups of Integers Modulo m Category:Cyclic Group of Order 3 b59mrv9uwib7stlw48ftagmhpq895h6


Alternating Group on 4 Letters/Subgroups 0 72772 388453 378472 2019-01-19T06:19:04Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Alternating Group on $4$ Letters == Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The subsets of $A_4$ which form subgroups of $A_4$ are as follows: Trivial: {{begin-eqn}} {{eqn | o =        | r = \set e       | c = Trivial Subgroup is Subgroup }} {{eqn | o =        | r = A_4       | c = Group is Subgroup of Itself }} {{end-eqn}} Order $2$ subgroups: {{begin-eqn}} {{eqn | o =        | r = \set {e, t}       | c = as $t^2 = e$ }} {{eqn | o =        | r = \set {e, u}       | c = as $u^2 = e$ }} {{eqn | o =        | r = \set {e, v}       | c = as $v^2 = e$ }} {{end-eqn}} Order $3$ subgroups: {{begin-eqn}} {{eqn | o =        | r = \set {e, a, p}       | c = as $a^2 = p$, $a^3 = a p = e$ }} {{eqn | o =        | r = \set {e, b, s}       | c = as $b^2 = s$, $b^3 = b s = e$ }} {{eqn | o =        | r = \set {e, c, q}       | c = as $c^2 = q$, $c^3 = c q = e$ }} {{eqn | o =        | r = \set {e, d, r}       | c = as $d^2 = r$, $d^3 = d r = e$ }} {{end-eqn}} Order $4$ subgroup: {{begin-eqn}} {{eqn | o =        | r = \set {e, t, u, v}       | c = Klein $4$-Group }} {{end-eqn}} </onlyinclude> == Examples of Subgroups == {{:Alternating Group on 4 Letters/Subgroups/Examples}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Alternating Group on 4 Letters/Order of Elements|next = Group does not Necessarily have Subgroup of Order of Divisor of its Order/Proof 2}}: Chapter $9$: Permutations: Exercise $3$ Category:Alternating Group on 4 Letters fcb3cssioaoq68bpr5xonor2rjqx8ny


Alternating Group on 4 Letters/Normality of Subgroups 0 72794 387204 385380 2019-01-11T14:09:46Z Prime.mover 59 wikitext text/x-wiki == Normality of Subgroups of the Alternating Group on $4$ Letters == Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The normality status of the non-trivial proper subgroups of $A_4$ is as follows: Order $2$ subgroups: {{begin-eqn}} {{eqn | l = T       | o = :=       | r = \set {e, t}       | c = Not normal }} {{eqn | l = U       | o = :=       | r = \set {e, u}       | c = Not normal }} {{eqn | l = V       | o = :=       | r = \set {e, v}       | c = Not normal }} {{end-eqn}} Order $3$ subgroups: {{begin-eqn}} {{eqn | l = P       | o = :=       | r = \set {e, a, p}       | c = Not normal }} {{eqn | l = Q       | o = :=       | r = \set {e, c, q}       | c = Not normal }} {{eqn | l = R       | o = :=       | r = \set {e, d, r}       | c = Not normal }} {{eqn | l = S       | o = :=       | r = \set {e, b, s}       | c = Not normal }} {{end-eqn}} Order $4$ subgroup: {{begin-eqn}} {{eqn | l = K       | o = :=       | r = \set {e, t, u, v}       | c = Normal }} {{end-eqn}} </onlyinclude> == Proof == Testing one of the left cosets of $T = \set {e, t}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = a T       | r = \set {a \circ e, a \circ t}       | c =  }} {{eqn | r = \set {a, c}       | c =  }} {{eqn | l = T a       | r = \set {e \circ a, t \circ a}       | c =  }} {{eqn | r = \set {a, b}       | c =  }} {{eqn | o = \ne       | r = a T       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $T$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $U = \set {e, u}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = b U       | r = \set {b \circ e, b \circ u}       | c =  }} {{eqn | r = \set {b, c}       | c =  }} {{eqn | l = U b       | r = \set {e \circ b, u \circ b}       | c =  }} {{eqn | r = \set {b, d}       | c =  }} {{eqn | o = \ne       | r = b U       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $U$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $V = \set {e, v}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = c V       | r = \set {c \circ e, c \circ v}       | c =  }} {{eqn | r = \set {c, d}       | c =  }} {{eqn | l = V c       | r = \set {e \circ c, v \circ c}       | c =  }} {{eqn | r = \set {c, b}       | c =  }} {{eqn | o = \ne       | r = c V       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $V$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $P = \set {e, a, p}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t P       | r = \set {t \circ e, t \circ a, t \circ p}       | c =  }} {{eqn | r = \set {t, b, q}       | c =  }} {{eqn | l = P t       | r = \set {e \circ t, a \circ t, p \circ t}       | c =  }} {{eqn | r = \set {t, c, s}       | c =  }} {{eqn | o = \ne       | r = t P       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $P$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $Q = \set {e, c, q}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t Q       | r = \set {t \circ e, t \circ c, t \circ q}       | c =  }} {{eqn | r = \set {t, d, p}       | c =  }} {{eqn | l = Q t       | r = \set {e \circ t, c \circ t, q \circ t}       | c =  }} {{eqn | r = \set {t, a, r}       | c =  }} {{eqn | o = \ne       | r = t Q       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $Q$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $R = \set {e, d, r}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t R       | r = \set {t \circ e, t \circ d, t \circ r}       | c =  }} {{eqn | r = \set {t, c, s}       | c =  }} {{eqn | l = R t       | r = \set {e \circ t, d \circ t, r \circ t}       | c =  }} {{eqn | r = \set {t, b, q}       | c =  }} {{eqn | o = \ne       | r = t R       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $R$ is not normal in $A_4$. {{qed|lemma}} Testing one of the left cosets of $S = \set {e, b, s}$ against its corresponding right coset: {{begin-eqn}} {{eqn | l = t S       | r = \set {t \circ e, t \circ b, t \circ s}       | c =  }} {{eqn | r = \set {t, a, r}       | c =  }} {{eqn | l = S t       | r = \set {e \circ t, b \circ t, s \circ t}       | c =  }} {{eqn | r = \set {t, d, p}       | c =  }} {{eqn | o = \ne       | r = t S       | c =  }} {{end-eqn}} The left coset does not equal the right coset and so $S$ is not normal in $A_4$. {{qed|lemma}} The cosets of $K = \set {e, t, u, v}$ are as follows: {{begin-eqn}} {{eqn | l = a K       | r = \set {a \circ e, a \circ t, a \circ u, a \circ v}       | c =  }} {{eqn | r = \set {a, c, d, b}       | c =  }} {{eqn | l = K a       | r = \set {e \circ a, t \circ a, u \circ a, v \circ a}       | c =  }} {{eqn | r = \set {a, b, c, d}       | c =  }} {{eqn | r = a K       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = p K       | r = \set {p \circ e, p \circ t, p \circ u, p \circ v}       | c =  }} {{eqn | r = \set {p, s, q, r}       | c =  }} {{eqn | l = K p       | r = \set {e \circ p, t \circ p, u \circ p, v \circ p}       | c =  }} {{eqn | r = \set {p, q, r, s}       | c =  }} {{eqn | r = p K       | c =  }} {{end-eqn}} The left cosets equal the right cosets and so $K$ is normal in $A_4$. {{qed|lemma}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Stabilizer of Polynomial|next = Normality Relation is not Transitive/Proof 1}}: Chapter $6$: Cosets: Exercise $11$ Category:Alternating Group on 4 Letters bichruyi89pkryb6htqxbl6dk2nznh0


Dihedral Group D4/Matrix Representation/Formulation 1 0 72869 389014 385553 2019-01-22T07:52:21Z Prime.mover 59 wikitext text/x-wiki == Matrix Representation of Dihedral Group $D_4$ == <onlyinclude> Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C$ denote the following four elements of the matrix space $\map {\mathcal M_\Z} 2$: :$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ The set: :$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$ under the operation of conventional matrix multiplication, forms the '''dihedral group $D_4$'''. </onlyinclude> === Cayley Table === Its Cayley table is given by: {{:Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table}} == Also see == * Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties. == Sources == {{SourceReview}} Category:Dihedral Group D4 fotfa45owyoai8vixxeyk8z782wx8gs


Dihedral Group D4/Matrix Representation/Formulation 1/Cayley Table 0 72870 385551 385533 2018-12-31T09:57:02Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$: :$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$ under the operation of conventional matrix multiplication, where: :$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ can be presented as: <onlyinclude> :$\begin{array}{r|rrrrrrrr}            &  \mathbf I &  \mathbf A &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\ \hline  \mathbf I &  \mathbf I &  \mathbf A &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\  \mathbf A &  \mathbf A &  \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I &  \mathbf C &  \mathbf B \\  \mathbf B &  \mathbf B &  \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C &  \mathbf I &  \mathbf A \\  \mathbf C &  \mathbf C &  \mathbf B &  \mathbf A &  \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I \\ -\mathbf I & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C &  \mathbf I &  \mathbf A &  \mathbf B &  \mathbf C \\ -\mathbf A & -\mathbf A & -\mathbf I &  \mathbf C &  \mathbf B &  \mathbf A &  \mathbf I & -\mathbf C & -\mathbf B \\ -\mathbf B & -\mathbf B & -\mathbf C &  \mathbf I &  \mathbf A &  \mathbf B & -\mathbf C & -\mathbf I & -\mathbf A \\ -\mathbf C & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I &  \mathbf C &  \mathbf B &  \mathbf A &  \mathbf I \end{array}$ </onlyinclude> == Sources == {{SourceReview}} Category:Dihedral Group D4 Category:Examples of Cayley Tables f7uc1dmd4ox2xz595bc5jvw9rsw7tsk


Quaternion Group/Group Presentation

Quaternion Group/Group Presentation 0 72932 445965 387854 2020-02-03T12:02:51Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Quaternion Group == The group presentation of the quaternion group is given by: <onlyinclude> :$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ </onlyinclude> == Proof == Let $G = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$. It is to be demonstrated that $\Dic 2$ is isomorphic to $G$. Consider the Cayley table for $\Dic 2$: {{:Quaternion Group/Cayley Table}} We have that: :$a^4 = e$ :$b^2 = a^2$ :$\paren {a b} a = b$ demonstrating that $\Dic 2$ has the same group presentation as $G$. Hence the result. {{qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/B, F|next = Definition:Left Coset}}: Chapter $4$: Subgroups: Exercise $4$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Generator of Group|next = Definition:Cyclic Group/Definition 2|entry = generator|index = 2}} Category:Quaternion Group Category:Group Presentations gyoyxuok1r5eambe6mx37emrp4yjam8


Bound for Analytic Function and Derivatives/Analytic Function Bounded on Circle 0 72944 413234 379993 2019-07-17T18:36:53Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f$ be a complex function. Let $z_0$ be a point in $\C$. Let $\Gamma$ be a circle in $\C$ with center at $z_0$ and radius in $\R_{>0}$. Let $f$ be analytic on $\Gamma$. Then $f$ is bounded on $\Gamma$. </onlyinclude> == Proof == Let: :$\map {f_{\Re} } z = \map \Re {\map f z}$ :$\map {f_{\Im} } z = \map \Im {\map f z}$ Let $\closedint a b$, $a < b$, be a closed real interval. Let $p$ be a continuous complex-valued function defined such that: :$\Gamma = \set {\map p u: u \in \closedint a b}$ $f$ is continuous on $\Gamma$ as $f$ is analytic on $\Gamma$ by the definition of analytic. Also, Real and Imaginary Part Projections are Continuous. Therefore, $f_{\Re}$ and $f_{\Im}$ are continuous by the corollary to Composite of Continuous Mappings is Continuous. Observe that $f_{\Re}$ and $f_{\Im}$ are real-valued functions that are continuous. Also, $p$ is a continuous function defined on a set of real numbers. Therefore $\map {f_{\Re} } {\map p u}$ and $\map {f_{\Im} } {\map p u}$ are continuous real functions by the corollary to Composite of Continuous Mappings is Continuous. $\map {f_{\Re} } {\map p u}$ and $\map {f_{\Im} } {\map p u}$ are bounded on $\closedint a b$ by Continuous Real Function is Bounded. Therefore $\map f {\map p u}$ is bounded on $\closedint a b$ as $\map f {\map p u} = \map {f_{\Re} } {\map p u} + i \map {f_{\Im} } {\map p u}$ where $i = \sqrt {-1}$. Accordingly, $f$ is bounded on $\Gamma$ as $\Gamma = \set {\map p u: u \in \closedint a b}$. {{qed}} Category:Complex Analysis gokqfygmi1t231kgrg5l4ovtp6573nv


Convergence of Taylor Series of Function Analytic on Disk/Lemma 0 72959 435251 380103 2019-11-14T15:05:50Z Prime.mover 59 wikitext text/x-wiki {{refactor|This (or something very similar) already exists somewhere as a basic null sequence}} == Lemma == <onlyinclude> Let $y > 1$. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac n {y^n} = 0$ </onlyinclude> == Proof == Note that $\ln y > 0$ as $y > 1$. {{begin-eqn}} {{eqn | l = \lim_{n \mathop \to \infty} \frac n {y^n}       | r = \lim_{n \mathop \to \infty} \frac n {\left({e^{\ln y} }\right)^n} }} {{eqn | r = \lim_{n \mathop \to \infty} \frac n {e^{\left({\ln y}\right) n} } }} {{eqn | r = 0       | c = as $\displaystyle \lim_{x \mathop \to \infty} \frac x {e^{\left({\ln y}\right) x} } = 0$ by Limit at Infinity of Polynomial over Complex Exponential as $\ln y > 0$ }} {{end-eqn}} {{qed}} Category:Real Analysis 00nqt72unqz24foiob6y1l26q9rmv46


Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma 0 72976 437673 435381 2019-12-03T22:41:36Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega_f} x = \displaystyle \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\map {\omega_f} I = \displaystyle \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ Let $\map {\omega^L_f} x \in \R$. Let $\map {\omega_f} x \in \R$. Then $\map {\omega^L_f} x = \map {\omega_f} x$. </onlyinclude> == Proof == We know that $\map {\omega^L_f} x$ and $\map {\omega_f} x$ are real numbers. We need to prove that $\map {\omega^L_f} x = \map {\omega_f} x$. Let $\epsilon \in \R_{>0}$. First, we aim to prove that $\size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x} < \epsilon$ for a small enough $h \in R_{>0}$. $\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ means by the definition of limit from the right that a strictly positive real number $h_1$ exists such that: :$\size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega^L_f} x)} < \epsilon$ for every $h$ that satisfies: $0 < h < h_1$. This means in particular that $\map {\omega_f} {\openint {x - h} {x + h} } \in \R$ for every $h$ that satisfies: $0 < h < h_1$. Let $h'$ be a real number that satisfies: $0 < h' < h_1$. We observe that $\openint {x - h'} {x + h'} \in N_x$. Therefore, $\map {\omega_f} {\openint {x - h'} {x + h'} } \in \set {\map {\omega_f} I: I \in N_x}$. By definition, $\map {\omega_f} x$ is a lower bound for $\set {\map {\omega_f} I: I \in N_x}$. Accordingly: :$\map {\omega_f} {\openint {x - h'} {x + h'} } \ge \map {\omega_f} x$ The fact that $\map {\omega_f} x \in \R$ implies that: :$\map {\omega_f} I - \map {\omega_f} x < \epsilon$ by Infimum of Set of Oscillations on Set is Arbitrarily Close for an $I \in N_x$. Let $I$ be such an element of $N_x$. We observe in particular that $\map {\omega_f} I \in \R$. A neighborhood in $N_x$ contains an open subset that contains the point $x$. So, $I$ contains such an open subset as $I \in N_x$. Therefore, a number $h_2 \in \R_{>0}$ exists such that $\openint {x - h_2} {x + h_2}$ is a subset of $I$. Let $h''$ be a real number that satisfies: $0 < h'' < h_2$. We observe that $\openint {x - h''} {x + h''}$ is a subset of $I$. We have: :$\map {\omega_f} I \in \R$ :$\openint {x - h''} {x + h''}$ is a subset of $I$ Therefore: :$\map {\omega_f} {\openint {x - h''} {x + h''} } \le \map {\omega_f} I$ by Oscillation on Subset Putting all this together, we get for every $h$ that satisfies: $0 < h < \min \set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I       | c = as $\map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }$ is true }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = \le       | r = \map {\omega_f} I < \map {\omega_f} x + \epsilon       | c = as $\map {\omega_f} I < \map {\omega_f} x + \epsilon$ is true }} {{eqn | ll= \leadsto       | l = \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }       | o = <       | r = \map {\omega_f} x + \epsilon       | c =  }} {{eqn | ll= \leadsto       | l = 0 \le \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x       | o = <       | r = \epsilon }} {{eqn | ll= \leadsto       | l = \size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x}       | o = <       | r = \epsilon }} {{end-eqn}} Thus, we achieved our first aim. Next, we get for every $h$ that satisfies: $0 < h < \min \set {h_1, h_2}$: {{begin-eqn}} {{eqn | l = \size {\map {\omega^L_f} x - \map {\omega_f} x}       | r = \size {\map {\omega^L_f} x - \map {\omega_f} {\openint {x - h} {x + h} } + \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x} }} {{eqn | o = \le       | r = \size {\map {\omega^L_f} x - \map {\omega_f} {\openint {x - h} {x + h} } } + \size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x}       | c = Triangle Inequality for Real Numbers }} {{eqn | o = <       | r = \epsilon + \epsilon }} {{eqn | r = 2 \epsilon }} {{end-eqn}} This holds for every $\epsilon \in \R_{>0}$. Therefore, $\size {\map {\omega^L_f} x - \map {\omega_f} x} = 0$ as $\size {\map {\omega^L_f} x - \map {\omega_f} x}$ is independent of $\epsilon$. Accordingly: :$\map {\omega^L_f} x = \map {\omega_f} x$ {{qed}} {{Improve|Implement a better category than this}} Category:Real Analysis 3pmygvx7f48jhe9eidgjxzeyt2q2wiy


Sum of Terms of Magic Cube/Sequence

Sum of Terms of Magic Cube/Sequence 0 73169 433513 382796 2019-11-01T14:01:22Z Prime.mover 59 wikitext text/x-wiki == Sequence of Sums of Terms of Magic Cubes == <onlyinclude> The sequence of the sum totals of all the entries in magic cubes of order $n$ begins: :$1, \paren {36,} \, 378, 2080, 7875, 23 \, 436, 58 \, 996, 131 \, 328, \ldots$ However, note that while $36 = \dfrac {2^3 \paren {2^3 + 1} } 2$, a magic cube of order $2$ does not actually exist. </onlyinclude> {{OEIS|A037270}} Category:Magic Cubes 2y8fzpedt6pmmf8b0svb3wy0m1xikwm


Magic Constant of Magic Cube/Sequence

Magic Constant of Magic Cube/Sequence 0 73171 433415 382868 2019-11-01T12:58:11Z Prime.mover 59 wikitext text/x-wiki == Sequence of Magic Constants of Magic Cubes == <onlyinclude> The sequence of the magic constants of magic cubes of order $n$ begins: :$1, (9,) \, 42, 130, 315, 651, 1204, 2052, 3285, 5005, 7326, 10 \, 374, 14 \, 287, 19 \, 215, 25 \, 320, 32 \, 776, \ldots$ However, note that while $9 = \dfrac {2 \paren {2^3 + 1} } 2$, a magic cube of order $2$ does not actually exist. </onlyinclude> {{OEIS|A027441}} Category:Magic Cubes 9gyu6xzac1eu8s55wz15feg80loz1xg


Sequence Lemma

Sequence Lemma 0 73179 490698 490697 2020-09-25T16:19:10Z Prime.mover 59 wikitext text/x-wiki {{tidy}} {{MissingLinks}} == Lemma == Let $A$ be a subset of a topological space $X$. If there is a sequence of points of $A$ converging to $x$, then $x \in \bar A$. The converse holds if $X$ is first-countable. == Proof == Assume the sequence of points of $A$ that converges to $x$ is $\sequence {x_i}$. Then for any open set $U$ of $x$, there exists a positive natural number $N$ such that when $i > N$, $x_i \in U$. Thus $U \cap A$ is nonempty, $x \in \bar A$. Let the topological space $X$ be first-countable. Then there is a countable collection of open neighbourhood $\family {U_i}_{i \mathop \in \Bbb Z_+}$ of $x$ such that any open neighbourhood $U$ of $x$ contains at least one of the sets $U_i$. Because $x \in \bar A$, $U_1 \cap A$ is nonempty, we can select a point $x_1$ in it. In a similar manner, $U_1 \cap U_2 \cap A$ is nonempty. Hence we can select a point $x_2$ in it. The point $x_i$ is selected from: :$U_1 \cap U_2 \cap \cdots \cap U_i \cap A$ We then obtain a sequence $\sequence {x_i}$. For any open neighbourhood $U$ of $x$, it contains at least one of the set $U_N$, $N \in \Bbb Z_+$ of $\family {U_i}_{i \mathop \in \Bbb Z_+}$. Thus it contains the set: :$U_1 \cap U_2 \cap \cdots \cap U_N \cap A$ :$U_1 \cap U_2 \cap \cdots \cap U_N \cap U_{N + 1}\cap A$ :$U_1 \cap U_2 \cap \cdots \cap U_N \cap U_{N + 1} \cap U_{N + 2} \cap A$ :$\ldots$ or the set $U_1 \cap U_2 \cap \cdots \cap U_i \cap A$ with $i > N$, hence the points $x_i$ with $i > N$. The sequence $\sequence {x_i}$ converges to $x$. {{qed}} == Sources == * {{BookReference|Topology|2004|James R. Munkres}} P130, Lemma 21.2 Category:Topology Category:Countability Axioms Category:First-Countable Spaces ta4s3r4pyrg9tkz7apjkpwtszep4f0q


Group Direct Product/Examples/C2 x C2

Group Direct Product/Examples/C2 x C2 0 73396 389230 386523 2019-01-23T15:31:00Z Prime.mover 59 wikitext text/x-wiki == Example of Group Direct Product == <onlyinclude> The direct product of $C_2$, the cyclic group of order $2$, with itself is as follows. Let us represent $C_2$ as the group $\struct {\set {1, -1}, \times}$: :$\begin {array} {r|rr} \struct {\set {1, -1} , \times} & 1 & -1 \\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array}$ Then the Cayley table for $C_2 \times C_2$ can be portrayed as: :$\begin {array} {c|cccc} C_2 \times C_2  & \tuple { 1,  1} & \tuple { 1, -1} & \tuple {-1,  1} & \tuple {-1, -1} \\ \hline \tuple { 1,  1} & \tuple { 1,  1} & \tuple { 1, -1} & \tuple {-1,  1} & \tuple {-1, -1} \\ \tuple { 1, -1} & \tuple { 1, -1} & \tuple { 1,  1} & \tuple {-1, -1} & \tuple {-1,  1} \\ \tuple {-1,  1} & \tuple {-1,  1} & \tuple {-1, -1} & \tuple { 1,  1} & \tuple { 1, -1} \\ \tuple {-1, -1} & \tuple {-1, -1} & \tuple {-1,  1} & \tuple { 1, -1} & \tuple { 1,  1} \\ \end{array}$ </onlyinclude> This is seen by inspection to be an instance of the Klein $4$-group. == Subgroups == {{:Group Direct Product/Examples/C2 x C2/Subgroups}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = External Direct Product of Groups is Group|next = Definition:Multiplicative Notation}}: Chapter $1$: Definitions and Examples: Example $1.10$ Category:Examples of Group Direct Products Category:Klein Four-Group prtc0kzrbj0gjde092r3sgogxcwrelf


Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table 0 73423 385557 385555 2018-12-31T12:05:18Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$: :$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$ under the operation of conventional matrix multiplication, where: :$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$ :$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \qquad \mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$ can be presented as: <onlyinclude> :$\begin{array}{r|rrrrrrrr}           & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \hline \mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \mathbf A & \mathbf A & \mathbf B & \mathbf C & \mathbf I & \mathbf E & \mathbf F & \mathbf G & \mathbf D \\ \mathbf B & \mathbf B & \mathbf C & \mathbf I & \mathbf A & \mathbf F & \mathbf G & \mathbf D & \mathbf E \\ \mathbf C & \mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf G & \mathbf D & \mathbf E & \mathbf F \\ \mathbf D & \mathbf D & \mathbf G & \mathbf F & \mathbf E & \mathbf I & \mathbf C & \mathbf B & \mathbf A \\ \mathbf E & \mathbf E & \mathbf D & \mathbf G & \mathbf F & \mathbf A & \mathbf I & \mathbf C & \mathbf B \\ \mathbf F & \mathbf F & \mathbf E & \mathbf D & \mathbf G & \mathbf B & \mathbf A & \mathbf I & \mathbf C \\ \mathbf G & \mathbf G & \mathbf F & \mathbf E & \mathbf D & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$ </onlyinclude> == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Square Matrices with +1 or -1 Determinant under Multiplication forms Group|next = Symmetry Group of Square/Cayley Table}}: Chapter $1$: Definitions and Examples: Exercise $2$ Category:Dihedral Group D4 Category:Examples of Cayley Tables guk2cvj7qlekm5o2z8a17vgul9mj7g8


Dihedral Group D4/Group Presentation

Dihedral Group D4/Group Presentation 0 73538 386381 386233 2019-01-05T08:46:01Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Dihedral Group $D_4$ == The group presentation of the dihedral group $D_4$ is given by: <onlyinclude> :$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$ </onlyinclude> == Proof == We have that the group presentation of the dihedral group $D_n$ is: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Setting $n = 4, \alpha = a, \beta = b$, we get: :$D_4 = \gen {a, b: a^4 = b^2 = e, b a b = a^{−1} }$ from which the result follows. {{qed}} Category:Dihedral Group D4 oh3atztie8viahgrxt735dwnqryv7xg


Dihedral Group D3/Group Presentation

Dihedral Group D3/Group Presentation 0 73560 386380 386377 2019-01-05T08:45:49Z Prime.mover 59 wikitext text/x-wiki == Group Presentation of Dihedral Group $D_3$ == The group presentation of the dihedral group $D_3$ is given by: <onlyinclude> :$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$ </onlyinclude> == Proof == We have that the group presentation of the dihedral group $D_n$ is: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Setting $n = 3, \alpha = a, \beta = b$, we get: :$D_3 = \gen {a, b: a^3 = b^2 = e, b a b = a^{−1} }$ from which the result follows. {{qed}} Category:Dihedral Group D3 675nb0ghqc462n8hcjps9rwo03tf0nt


Dihedral Group D3/Cayley Table

Dihedral Group D3/Cayley Table 0 73563 386386 2019-01-05T09:08:29Z Prime.mover 59 Created page with "== Cayley Table of Dihedral Group $D_3$ ==  The Cayley table of the Definition:Dihed..." wikitext text/x-wiki == Cayley Table of Dihedral Group $D_3$ == The Cayley table of the dihedral group $D_3$ can be written: <onlyinclude> :$\begin{array}{c|cccccc}       & e     & a     & a^2   & b     & a b   & a^2 b \\ \hline e     & e     & a     & a^2   & b     & a b   & a^2 b \\ a     & a     & a^2   & e     & a b   & a^2 b & b     \\ a^2   & a^2   & e     & a     & a^2 b & b     & a b   \\ b     & b     & a^2 b & a b   & e     & a^2   & a     \\ a b   & a b   & b     & a^2 b & a     & e     & a^2   \\ a^2 b & a^2 b & a b   & b     & a^2   & a     & e     \\ \end{array}$ </onlyinclude> where the group presentation of $D_3$ is given as: {{:Group Presentation of Dihedral Group D3}} Category:Examples of Cayley Tables Category:Dihedral Group D3 89aknx2cuypw4ge7xzprwq9gksdrk06


Standard Parity Check Matrix/Examples/(6, 3) code in Z2 0 73781 387552 387549 2019-01-13T10:35:07Z Prime.mover 59 wikitext text/x-wiki == Example of Standard Parity Check Matrix == <onlyinclude> Let $C$ be the linear $\tuple {6, 3}$-code in $\Z_2$ whose standard generator matrix $G$ is given by: :$G := \begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 0 \\  0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}$ Its standard parity check matrix $P$ is given by: :$P := \begin{pmatrix} 1 & 1 & 0 & 1 & 0 & 0 \\  1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{pmatrix}$ </onlyinclude> == Proof == Expressing $G$ in the form: :$G = \paren {\begin{array} {c|c} \mathbf I_k & \mathbf A \end{array} }$ it is seen that: :$\mathbf A = \begin{pmatrix} 1 & 1 & 0 \\  1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ It is noted that $\mathbf A^\intercal$ is: :$\mathbf A^\intercal =  \begin{pmatrix} 1 & 1 & 0 \\  1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$ as $\mathbf A$ is symmetrical about the main diagonal. Then each of the elements of $\Z_2$ is self-inverse, so: :$-\mathbf A^\intercal = \mathbf A^\intercal$ {{Qed}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Definition:Standard Parity Check Matrix|next = Definition:Syndrome}}: Chapter $6$: Error-correcting codes: Example $6.18$ Category:Examples of Linear Codes qr1jfa03pvbul95tdgxdv9khqnvkcg1


Syndrome Decoding/Examples/(6, 3) code in Z2 0 73802 387561 2019-01-13T13:16:59Z Prime.mover 59 Created page with "== Example of Syndrome Decoding == <onlyinclude> Let $C$ be the linear code:  :$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110..." wikitext text/x-wiki == Example of Syndrome Decoding == <onlyinclude> Let $C$ be the linear code: :$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$ Then the Syndrome Decoding table $T$ for $C$ is: :$\begin{array} {cc} 000000 & 000 \\ 100000 & 110 \\ 010000 & 101 \\ 001000 & 011 \\ 000100 & 100 \\ 000010 & 010 \\ 000001 & 001 \\ 100001 & 111 \\ \end{array}$ </onlyinclude> To find the $8$th row, it is not necessary to hunt directly an element $v$ of $\map V {6, 2}$ of weight $2$ which does not exist anywhere in the other $7$ rows. What you do is identify the remaining syndrome of $3$ digits that has not been used yet (that is: $111$). Then you work out what combinations of coset leaders and their own syndromes which when added together make that last syndrome. The fact that in this case that combination is not unique (here we get $100001$, $010010$, $001100$ and $000111$) means that an element of $\map V {6, 2}$ which is more than $2$ distant from a codeword cannot be uniquely decoded. == Example == {{:Syndrome Decoding/Examples/(6, 3) code in Z2/Example}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Syndrome Decoding|next = Syndrome Decoding/Examples/(6, 3) code in Z2/Example}}: Chapter $6$: Error-correcting codes: Example $6.22$ Category:Examples of Linear Codes qdysn310ovs76bzkwix1spvm2ut3zse


Dihedral Group D4/Cayley Table

Dihedral Group D4/Cayley Table 0 73840 387808 387779 2019-01-15T07:52:51Z Prime.mover 59 wikitext text/x-wiki == Cayley Table for Dihedral Group $D_4$ == The Cayley table for the dihedral group $D_4$, whose group presentation is: {{:Group Presentation of Dihedral Group D4}} can be presented as: <onlyinclude> :$\begin{array}{l|cccccccc}        &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\ \hline  e     &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\  a     &     a &   a^2 &   a^3 &     e & b a^3 &     b &   b a & b a^2 \\  a^2   &   a^2 &   a^3 &     e &     a & b a^2 & b a^3 &     b &   b a \\  a^3   &   a^3 &     e &     a &   a^2 &   b a & b a^2 & b a^3 &     b \\  b     &     b &   b a & b a^2 & b a^3 &     e &     a &   a^2 &   a^3 \\  b a   &   b a & b a^2 & b a^3 &     b &   a^3 &     e &     a &   a^2 \\  b a^2 & b a^2 & b a^3 &     b &   b a &   a^2 &   a^3 &     e &     a \\  b a^3 & b a^3 &     b &   b a & b a^2 &     a &   a^2 &   a^3 &     e \end{array}$ </onlyinclude> === Coset Decomposition of $\set {e, a^2}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen a^2$ gives: {{:Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)}} === Coset Decomposition of $\set {e, a, a^2, a^3}$ === Presenting the above Cayley table with respect to the coset decomposition of the normal subgroup $\gen a$ gives: {{:Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a, a^2, a^3)}} == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Dihedral Group D4/Normal Subgroups/Subgroup Generated by a^2/Quotient Group|next = Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)}}: Chapter $7$: Normal subgroups and quotient groups: Example $7.13$ Category:Dihedral Group D4 Category:Examples of Cayley Tables lk9n3g9c3rc1tk0fyai15cfnc7c0q8i


Quaternion Group/Subgroups

Quaternion Group/Subgroups 0 73851 387863 387860 2019-01-15T11:20:53Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Quaternion Group == Let $Q$ denote the quaternion group, whose group presentation is given as: {{:Group Presentation of Quaternion Group}} <onlyinclude> The subsets of $Q$ which form subgroups of $Q$ are: {{begin-eqn}} {{eqn | o =        | r = Q }} {{eqn | o =        | r = \set e }} {{eqn | o =        | r = \set {e, a^2} }} {{eqn | o =        | r = \set {e, a, a^2, a^3} }} {{eqn | o =        | r = \set {e, b, a^2, a^2 b} }} {{eqn | o =        | r = \set {e, a b, a^2, a^3 b} }} {{end-eqn}} From Quaternion Group is Hamiltonian we have that all of these subgroups of $Q$ are normal. </onlyinclude> == Proof == Consider the Cayley table for $Q$: {{:Quaternion Group/Cayley Table}} We have that: :$a^4 = e$ and so $\gen a = \set {e, a, a^2, a^3}$ forms a subgroup of $Q$ which is cyclic. We have that: :$b^2 = a^2$ and so $\gen b = \set {e, b, a^2, a^2 b}$ forms a subgroup of $Q$ which is cyclic. We have that: :$\paren {a b}^2 = a^2$ and so $\gen {a b} = \set {e, a b, a^2, a^3 b}$ forms a subgroup of $Q$ which is cyclic. We have that: :$\paren {a^2}^2 = e$ and so $\gen {a^2} = \set {e, a^2}$ forms a subgroup of $Q$ which is also a subgroup of $\gen a$, $\gen b$ and $\gen {a b}$. That exhausts all elements of $Q$. Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above will generate the whole of $Q$. {{qed}} == Also see == * Quaternion Group is Hamiltonian == Sources == * {{BookReference|A Course in Group Theory|1996|John F. Humphreys|prev = Symmetric Group on 3 Letters/Normal Subgroups|next = Quaternion Group is Hamiltonian}}: Chapter $7$: Normal subgroups and quotient groups: Exercise $3$ Category:Quaternion Group 7liu2qz525m458f5b51dpwqwtyj0y46


Dihedral Group D4/Subgroups

Dihedral Group D4/Subgroups 0 73855 497569 497568 2020-11-02T16:20:32Z Prime.mover 59 wikitext text/x-wiki == Subgroups of the Dihedral Group $D_4$ == Let the dihedral group $D_4$ be represented by its group presentation: {{:Group Presentation of Dihedral Group D4}} <onlyinclude> The subsets of $D_4$ which form subgroups of $D_4$ are: {{begin-eqn}} {{eqn | o =        | r = D_4 }} {{eqn | o =        | r = \set e }} {{eqn | o =        | r = \set {e, a, a^2, a^3} }} {{eqn | o =        | r = \set {e, a^2} }} {{eqn | o =        | r = \set {e, b} }} {{eqn | o =        | r = \set {e, b a} }} {{eqn | o =        | r = \set {e, b a^2} }} {{eqn | o =        | r = \set {e, b a^3} }} {{eqn | o =        | r = \set {e, a^2, b, b a^2} }} {{eqn | o =        | r = \set {e, a^2, b a, b a^3} }} {{end-eqn}} </onlyinclude> == Proof == Consider the Cayley table for $D_4$: {{:Dihedral Group D4/Cayley Table}} We have that: :$a^4 = e$ and so $\gen a = \set {e, a, a^2, a^3}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\paren {a^2}^2 = e$ and so $\gen {a^2} = \set {e, a^2}$ forms a subgroup of $D_4$ which is cyclic, and also a subgroup of $\gen a$. We have that: :$b^2 = e$ and so $\gen b = \set {e, b}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\paren {b a}^2 = e$ and so $\gen {b a} = \set {e, b a}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\paren {b a^2}^2 = e$ and so $\gen {b a^2} = \set {e, b a^2}$ forms a subgroup of $D_4$ which is cyclic. We have that: :$\paren {b a^3}^2 = e$ and so $\gen {b a^3} = \set {e, b a^3}$ forms a subgroup of $D_4$ which is cyclic. Then we have that: :$b a^2 = a^2 b$ and so $\gen {a^2, b} = \set {e, a^2, b, b a^2}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\set {e, a^2}$, $\set {e, b}$, $\set {e, b a^2}$. Then we have that: :$b a^3 = a^2 b a$ and so $\gen {a^2, b a} = \set {e, a^2, b a, b a^3}$ forms a subgroup of $D_4$ which is not cyclic, but which has subgroups $\set {e, a^2}$, $\set {e, b}$, $\set {e, b a^2}$. That exhausts all elements of $D_4$. Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above generate the whole of $D^4$. {{qed}} Category:Dihedral Group D4 h8f00ilfqs8wjrsvc502y7dpox94vsu