The symmetric group 
of degree 
is the group of all permutations
on 
symbols, where 
is a nonnegative integer. Its group
operation is function composition, and
it is a permutation group of group
order 
.
In particular, 
and 
are trivial groups. By Cayley's
group theorem, every finite group of group
order 
is isomorphic to a subgroup of 
.
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
, n].
The number of conjugacy classes of 
is given by 
, where 
is the partition function
P of 
.
For 
,
the symmetric group is a transitive group (Holton
and Sheehan 1993, p. 27).
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 integers 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/Inline25.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 
, where 
is a positive integer,
is given by
 |
(1)
|
where the sum runs over vectors 
of nonnegative
integers satisfying
 |
(2)
|
(Harary 1994, p. 184). The cycle indices for the first few 
are
 |  |  |
(3)
| ||
 |  |  |
(4)
| ||
 |  |  |
(5)
| ||
 |  |  |
(6)
| ||
 |  |  |  |  |
(7)
|
Netto's conjecture states that the probability that two permutations chosen independently and uniformly
from 
,
with replacement, generate the entire group tends to 3/4
as 
.
This was proved by Dixon (1969). The probabilities
for 
,
2, ... are 1, 3/4, 1/2, 3/8, 19/40, 53/120, 103/168, ... (OEIS A040173
and A040174).
