Symmetric Group

The symmetric group of degree is the group of all permutations on symbols. is therefore a permutation group of order and contains as subgroups every group of order .

The th symmetric group is represented in the Wolfram Language as SymmetricGroup[n]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[n], x1, ..., xn].

The number of conjugacy classes of is given , where is the partition function P of . The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).

For any finite group , Cayley's group theorem proves is isomorphic to a subgroup of a symmetric group.

The multiplication table for is illustrated above.

Let be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for , which has elements.

	(1)(2)(3)	(1)(23)	(3)(12)	(123)	(132)	(2)(13)
(1)(2)(3)	(1)(2)(3)	(1)(23)	(3)(12)	(123)	(132)	(2)(13)
(1)(23)	(1)(23)	(1)(2)(3)	(132)	(2)(13)	(3)(12)	(123)
(3)(12)	(3)(12)	(123)	(1)(2)(3)	(1)(23)	(2)(13)	(132)
(123)	(123)	(3)(12)	(2)(13)	(132)	(1)(2)(3)	(1)(23)
(132)	(132)	(2)(13)	(1)(23)	(1)(2)(3)	(123)	(3)(12)
(2)(13)	(2)(13)	(132)	(123)	(3)(12)	(1)(23)	(1)(2)(3)

This may be somewhat clearer to understand by using a sequence of three integers to denote both a given permutation and the ordering of numbers after applying a permutation. For example, consider the sequence , and apply to it the permutation that places the terms of a sequence in the order . In the notation of the Wolfram Language, this then gives , which is the identity permutation, as indicated in the table below.

	123	132	213	231	312	321
123	123	132	213	231	312	321
132	132	123	312	321	213	231
213	213	231	123	132	321	312
231	231	213	321	312	123	132
312	312	321	132	123	231	213
321	321	312	231	213	132	123

The cycle index (in variables , ..., ) for the symmetric group is given by

(1)

(Harary 1994, p. 184), where the sum runs over the set of solution vectors to

(2)

The cycle indices for the first few are

			(3)
			(4)
			(5)
			(6)
			(7)

Netto's conjecture states that the probability that two elements and of a symmetric group generate the entire group tends to 3/4 as . This was proven by Dixon (1969). The probability that two elements generate for , 2, ... are 1, 3/4, 1/2, 3/8, 19/40, 53/120, 103/168, ... (OEIS A040173 and A040174). Finding a general formula for terms in the sequence is a famous unsolved problem in group theory.

Wolfram Web Resources