Symmetric group

Cayley graph of the symmetric group of index 4 (S4) represented as the group of rotations of standard dice.

Cayley graph of the symmetric group of index 4 (S₄) represented as the group of rotations of standard dice. In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French

In mathematics, the symmetric group on a set X, denoted by S_X or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a composite function represents the application of one function to the results of another e. , two such functions f and g can be composed to yield a new bijective function $f \circ g$ , defined by $(f \circ g)(x) = f(g(x))$ for all x in X. Using this operation, S_X forms a group. The operation is also written as fg (and sometimes, although not here, as gf).

1 Finite symmetric groups
2 Composition of permutations
- 2.1 Verification of group axioms
3 Transpositions
4 Cycles
5 Conjugacy classes
6 Examples
- 6.1 Elements
7 Automorphism group
8 See also

Finite symmetric groups

Of particular importance is the symmetric group on the finite set

X = {1, . . . , n},

denoted by S_n. The permutations of X form the set of bijective functions. In several fields of Mathematics the term permutation is used with different but closely related meanings The group S_n has order n! and is not abelian for n > 2. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Similarly, the group S_n is solvable if and only if n ≤ 4. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally ↔ The remainder of this article will discuss S_n.

Subgroups of S_n are called permutation groups. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation

Composition of permutations

The group operation in a symmetric group is function composition; the symbol $\circ$ is often omitted, is demonstrated below. In Mathematics, a composite function represents the application of one function to the results of another Let

$f = (1\ 3)(4\ 5)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix}$

and

$g = (1\ 2\ 5)(3\ 4)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}$ . (See permutation for an explanation of notation). In several fields of Mathematics the term permutation is used with different but closely related meanings

Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives

$fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{pmatrix}$ .

A cycle of length L =k·m, taken to the k-th power, will decompose into k cycles of length m: For example (k=2, m=3),

$(1~2~3~4~5~6)^2 = (1~3~5) (2~4~6) ~.$

Verification of group axioms

To check that the symmetric group on a set X is indeed a group, it is necessary to verify the group axioms of associativity, identity, and inverses. Let S be a set A cycle is a Permutation ( bijective function of a set onto itself such that there exist distinct elements a_1 a_2\ldotsa_k In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The operation of function composition is always associative. In Mathematics, a composite function represents the application of one function to the results of another The trivial bijection that assigns each element of X to itself serves as an identity for the group. Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B

Transpositions

A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 2)(1 5)(3 4). Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas f is an even permutation. In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even

The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd.

To see this, consider the function P which maps a permutation to the number of pairs $(i, j), i < j,$ for which $f (j) < f (i)$ . Intuitively, this is the total number of times each of the numbers $1,\ldots,n$ has to cross over another number to arrive at its final position; it is independent of the way in which the permutation is written down. In fact, the parity of $P (f)$ corresponds to the odd- or even-ness of the permutation $f$ , as we will now show. It is fairly straightforward to prove that for a transposition $(a~b)$ , $P(a~b) = 2(b-a-2)+1$ , which is odd. Furthermore, $P(f(a~b))$ has opposite parity to $P (f)$ since we must either add or subtract $P(a~b)$ . Thus if we write $f$ as a product $f_1 f_2 \ldots f_m$ of transpositions the parity of $P (f)$ is the same as the parity of $m$ . This shows that, given any two products of transpositions that yield the same permutation, their respective numbers of transpositions must have the same parity.

The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Thus we can define the sign of a permutation:

$\operatorname{sgn}(f)=\left\{\begin{matrix} +1, & \mbox{if }f\mbox { is even} \\ -1, & \mbox{if }f \mbox{ is odd}. \end{matrix}\right.$

With this definition,

sgn: S_n → {+1, –1}

is a group homomorphism ({+1, –1} is a group under multiplication, where +1 is e, the neutral element). In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The kernel of this homomorphism, i. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism e. the set of all even permutations, is called the alternating group A_n. In Mathematics, an alternating group is the group of Even permutations of a Finite set. It is a normal subgroup of S_n, and for n ≥ 2 it has n! / 2 elements. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. The group S_n is the semidirect product of A_n and any subgroup generated by a single transposition. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can

Cycles

A cycle is a permutation f for which there exists an element x in {1,. Let S be a set A cycle is a Permutation ( bijective function of a set onto itself such that there exist distinct elements a_1 a_2\ldotsa_k . . ,n} such that x, f(x), f²(x), . . . , f^k(x) = x are the only elements moved by f. The permutation h defined by

$h = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end{pmatrix}$

is a cycle, since h(1) = 4, h(4) = 3 and h(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3). The length of this cycle is three. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are disjoint if they move different elements. Disjoint cycles commute, e. g. in S₆ we have (3 1 4)(2 5 6) = (2 5 6)(3 1 4). Every element of S_n can be written as a product of disjoint cycles; this representation is unique up to the order of the factors. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

Conjugacy classes

The conjugacy classes of S_n correspond to the cycle structures of permutations; that is, two elements of S_n are conjugate in S_n if and only if they consist of the same number of disjoint cycles of the same lengths. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class For instance, in S₅, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S_n can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example:

$k = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5\end{pmatrix}$

Which can be written as the product of cycles, namely:

(2 4)

This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, i. e.

(2 4)(1 2 3)(4 5)(2 4)=(1 4 3)(2 5)

It is clear that such a permutation is not unique.

Examples

The group S₃ is isomorphic to the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Properties The area of an equilateral triangle with sides of length a\\! Cycles of length two correspond to reflections, and cycles of length three are rotations.

For a list of elements of S₄, see Cycle notation. In combinatorial Mathematics, the cycle notation is a useful convention for writing down a Permutation in terms of its constituent cycles See cube for the proper rotations of a cube, which form a group isomorphic with S₄. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. In this case, the permuted objects are the four diagonals of the cube.

Symmetric groups are Coxeter groups and reflection groups. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries A reflection group is a Group action, acting on a finite dimensional Vector space, which is generated by reflections elements that fix a Hyperplane in They can be realized as a group of reflections with respect to hyperplanes $x_i=x_j, 1\leq i < j \leq n$ . Braid groups B_n admit symmetric groups S_n as quotient groups. In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on the elements of G, as a group acts on itself faithfully by (left or right) multiplication. In Group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a Subgroup

Elements

Certain elements of the symmetric group are of particular interest.

The reverse permutation reverses the order of elements:

$\begin{pmatrix} 1 & 2 & \cdots & n\\ n & n-1 & \cdots & 1\end{pmatrix}$

This is the longest element with respect to the Bruhat order, thinking of the symmetric group as a Coxeter group generated by the adjacent transpositions. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries

This is an involution, and consists of $\lfloor n/2 \rfloor$ (non-adjacent) transpositions: $(1\,n)(2\,n-1)\cdots$ , or $\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2}$ adjacent transpositions: $(n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots$ , so it thus has sign:

$\mbox{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} =(-1)^{n(n-1)/2} = \begin{cases} +1 & n \equiv 0,1 \pmod{4}\\ -1 & n \equiv 2,3 \pmod{4} \end{cases}$

which is 4-periodic in n.

In $S 2 n$ , the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them. The faro shuffle is a method of Shuffling Playing cards In a perfect shuffle or perfect faro shuffle the deck is split into equal halves of 26 cards which are Its sign is also $(-1)^{\lfloor n/2 \rfloor}$ .

Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic. In Mathematics, Clifford algebras are a type of Associative algebra.

Automorphism group

For more details on this topic, see Automorphisms of the symmetric and alternating groups. In Group theory, a branch of Mathematics, the Automorphisms and Outer automorphisms of the Symmetric groups and Alternating groups are

n	$Aut(S n)$	$Out(S n)$
$n\neq 2,6$	$S n$	1
$n = 2$	1	1
$n = 6$	$S_6 \rtimes C_2$	$C 2$

For $n \neq 2, 6$ , $S n$ is a complete group: its automorphism group is itself: it has no center or outer automorphisms. In Mathematics, a group G is said to be complete if every Automorphism of G is inner, and the group is a centerless group In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner

For n = 2, the automorphism group is trivial, but $S 2$ is not trivial (it is isomorphic to $C 2$ , which is abelian).

For n = 6, it has an outer automorphism of order 2: $Out(S 6) = C 2$ , and the automorphism group is a semidirect product: $\mbox{Aut}(S_6)=S_6 \rtimes C_2$ .

Dictionary

-noun

(mathematics) A group whose elements are precisely all of the bijections of some set with itself and whose operation is composition of those bijections.

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