Endomorphism structure of symmetric groups

This article gives specific information, namely, endomorphism structure, about a family of groups, namely: symmetric group.
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

Automorphism structure

• For $n\neq 2,6$, the symmetric group of degree $n$ is a complete group, i.e., the action on itself by conjugation gives an isomorphism from the group to its automorphism group. In particular, the symmetric group is isomorphic to its automorphism group.
• For $n=2$, the automorphism group is trivial.
• For $n=6$, i.e., for symmetric group:S6, the action on itself by conjugation induces an injective homomorphism to the automorphism group that is not surjective. The image has index two in the automorphism group. In other words, the outer automorphism group is cyclic group:Z2. See automorphism group of symmetric group:S6.

Other endomorphisms

• For $n=0,1$, every endomorphism is an automorphism.
• For $n=2$, every endomorphism is either trivial or an automorphism.
• For $n\geq 3$ and $n\neq 4$, 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 $n\neq 0,1,2,4,6$, 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