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 ![{2,1,3}[[{2,1,3}]]={1,2,3}](/images/equations/SymmetricGroup/Inline23.svg)
, 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.
