# Endomorphism structure of symmetric groups

## Contents

View endomorphism structure of group families | View other specific information about symmetric group
$n$ $n!$ (order of symmetric group) symmetric group of degree $n$ endomorphism structure page
0 1 trivial group --
1 1 trivial group --
2 2 cyclic group:Z2 --
3 6 symmetric group:S3 endomorphism structure of symmetric group:S3
4 24 symmetric group:S4 endomorphism structure of symmetric group:S4
5 120 symmetric group:S5 endomorphism structure of symmetric group:S5
6 720 symmetric group:S6 endomorphism structure of symmetric group:S6

## Endomorphism structure

### Other endomorphisms

• For $n = 0,1$, every endomorphism is an automorphism.
• For $n = 2$, every endomorphism is either trivial or an automorphism.
• For $n \ge 3$ and $n \ne 4$, every endomorphism is of one of these three types: an automorphism, the trivial map, or a map with image a cyclic subgroup of order two, obtained by applying the sign homomorphism. For the last type of endomorphism, if the non-identity element in the image is an odd permutation, then the map is a retraction, whereas if it is an even permutation, it is not a retraction.
• For $n = 4$, there are additional endomorphisms possible due to the presence of a normal V4 in S4.

Thus, for a generic $n \ne 0,1,2,4,6$, we have:

Quantity Value
order of automorphism group $n!$ (it's the symmetric group itself)
order of endomorphism monoid $n! + 1$ plus the number of elements of $S_n$ of order two.
number of retractions $2$ plus the number of odd permutations of $S_n$ of order two.

### Summary information

$n$ $n!$ (order of symmetric group) symmetric group of degree $n$ number of odd permutations of order two total number of permutations of order two automorphism group (= symmetric group of degree $n$ except when $n = 2,6$) order of automorphism group (= $n!$ except when $n = 2,6$) order of outer automorphism group (= 1 except when $n = 6$) order of endomorphism monoid (= order of automorphism group + total number of permutations of order two + 1, except when $n = 0,1,2,4$) number of retractions (= 2 + number of odd permutations of order two, except when $n = 0,1,2,4$)
0 1 trivial group 0 0 trivial group 1 1 1 1
1 1 trivial group 0 0 trivial group 1 1 1 1
2 2 cyclic group:Z2 1 1 trivial group 1 1 2 2
3 6 symmetric group:S3 3 3 symmetric group:S3 6 1 10 5
4 24 symmetric group:S4 6 9 symmetric group:S4 24 1 58 12
5 120 symmetric group:S5 10 25 symmetric group:S5 120 1 146 12
6 720 symmetric group:S6 30 75 automorphism group of symmetric group:S6 1440 2 1516 32
7 5040 symmetric group:S7 126 231 symmetric group:S7 5040 1 5272 128