# Endomorphism structure of symmetric group:S4

## Contents

View element structure of particular groups | View other specific information about symmetric group:S4

This article the endomorphism structure of discusses symmetric group:S4, the symmetric group of degree four. We denote its elements as acting on the set $\{ 1,2,3,4 \}$, written using cycle decompositions, with composition by function composition where functions act on the left.

## Summary of information

Construct Value Order Second part of GAP ID (if group)
endomorphism monoid  ? 58 --
automorphism group symmetric group:S4 24 12
inner automorphism group symmetric group:S4 24 12
extended automorphism group direct product of S4 and Z2 48 48

## Family contexts

Family name Parameter values General discussion of endomorphism structure of family
symmetric group 4 endomorphism structure of symmetric groups
projective general linear group of degree two field:F3 endomorphism structure of projective general linear group of degree two over a finite field

## Description of automorphism group

Symmetric group:S4 is a complete group (i.e., it is a centerless group and every automorphism is inner). See also symmetric groups on finite sets are complete.

Thus, all its automorphisms are inner automorphisms, i.e., they are given as conjugations by elements of the group, and distinct elements give distinct inner automorphisms.

## Other endomorphisms

### Summary

Note that the first row here gives the automorphisms.

Kernel of endomorphism Quotient by kernel (isomorphic to image) Possibilities for image Number of possible kernels Number of possible images Size of automorphism group of quotient Number of endomorphisms (product of three preceding column values) Number of endomorphisms up to pre-composition by an automorphism Number of endomorphisms up to post-composition by an automorphism Number of endomorphisms up to pre- and post-composition by automorphisms Number of endomorphisms up to conjugation by an automorphism
trivial subgroup symmetric group:S4 the whole group 1 1 24 24 1 1 1 5
normal V4 in S4 symmetric group:S3 S3 in S4 (any of four conjugate subgroups) 1 4 6 24 4 1 1 3
A4 in S4 cyclic group:Z2 S2 in S4 (6 possible conjugate subgroups) or subgroup generated by double transposition in S4 (3 possible conjugate subgroups) 1 9 1 9 9 2 2 2
whole group trivial group trivial subgroup 1 1 1 1 1 1 1 1
Total -- -- -- -- -- 58 15 5 5 11

## GAP implementation

The endomorphism structure can be explored using the GAP function Endomorphisms, available through the SONATA package:

```gap> L := Endomorphisms(SymmetricGroup(4));
[ [ (1,2,3,4), (1,2) ] -> [ (), () ], [ (1,2,3,4), (1,2) ] -> [ (1,4), (1,4) ], [ (1,2,3,4), (1,2) ] -> [ (2,3), (2,3) ],
[ (1,2,3,4), (1,2) ] -> [ (3,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (2,4), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,4), (2,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,3), (2,3) ],
[ (1,2,3,4), (1,2) ] -> [ (2,4), (1,4) ], [ (1,2,3,4), (1,2) ] -> [ (2,3), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,2), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (3,4), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (2,4), (1,2) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (3,4), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (2,3), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (1,4), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,3), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (2,4), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (2,3), (2,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,4), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (3,4), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (2,3), (3,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,3), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (3,4), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (2,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,4,2), (2,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,3,2), (2,3) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2,3,4), (1,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,2,4,3), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,2,4), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2,3,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,2,3), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,3,2), (1,2) ],
[ (1,2,3,4), (1,2) ] -> [ (1,4,2,3), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,2,4,3), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,2,4), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3,4,2), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,4,2), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,3,2), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2,3,4), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,2,4,3), (2,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,2,3), (1,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2,4,3), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,2,4), (2,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,3,4,2), (3,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3,2,4), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,2,3,4), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (1,4,2,3), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,4,3,2), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,3)(2,4), (1,3)(2,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,4)(2,3), (1,4)(2,3) ],
[ (1,2,3,4), (1,2) ] -> [ (1,2)(3,4), (1,2)(3,4) ] ]
gap> Length(L);
58
gap> M := Filtered(L,x -> x = x*x);
[ [ (1,2,3,4), (1,2) ] -> [ (), () ], [ (1,2,3,4), (1,2) ] -> [ (1,4), (1,4) ], [ (1,2,3,4), (1,2) ] -> [ (2,3), (2,3) ],
[ (1,2,3,4), (1,2) ] -> [ (3,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (2,4), (2,4) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3), (1,3) ], [ (1,2,3,4), (1,2) ] -> [ (1,3), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (2,4), (1,2) ],
[ (1,2,3,4), (1,2) ] -> [ (1,3), (1,2) ], [ (1,2,3,4), (1,2) ] -> [ (2,4), (3,4) ], [ (1,2,3,4), (1,2) ] -> [ (1,2,3,4), (1,2) ] ]
gap> Length(M);
12
gap> K := List(L,Kernel);
[ Group([ (1,2,3,4), (1,2) ]), Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,3)(2,4), (1,4,3) ]),
Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]),
Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,3)(2,4), (1,4)(2,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group(()), Group(()), Group(()), Group(()),
Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()),
Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group([ (1,3)(2,4), (1,4,3) ]), Group([ (1,3)(2,4), (1,4,3) ]),
Group([ (1,3)(2,4), (1,4,3) ]) ]
gap> FrequencySort(K);
[ [ Group(()), 24 ], [ Group([ (1,2,3,4), (1,2) ]), 1 ], [ Group([ (1,3)(2,4), (1,4,3) ]), 9 ], [ Group([ (1,4)(2,3), (1,3)(2,4) ]), 24 ] ]```