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)
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(Harary 1994, p. 184), where the sum runs over the set of solution vectors to
(2)
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The cycle indices for the first few are
(3)
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(4)
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(5)
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(6)
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(7)
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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.