# Endomorphism structure of symmetric group:S5

View element structure of particular groups | View other specific information about symmetric group:S5

This article discusses symmetric group:S5, the symmetric group of degree five. We denote its elements as acting on the set $\{ 1,2,3,4,5 \}$, 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  ? 146 --
automorphism group symmetric group:S5 120 34
inner automorphism group symmetric group:S5 120 34
extended automorphism group direct product of S5 and Z2 240 189

## Family contexts

Family name Parameter values General discussion of endomorphism structure of family
symmetric group 5 endomorphism structure of symmetric groups
projective general linear group of degree two field:F5 endomorphism structure of projective general linear group of degree two over a finite field

## Description of automorphism group

Symmetric group:S5 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

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 retractions
trivial subgroup symmetric group:S5 the whole group 1 1 120 120 1
A5 in S5 cyclic group:Z2 S2 in S5 (10 possible conjugate subgroups), subgroup generated by double transposition in S5 (15 possible conjugate subgroups) 1 25 1 25 10
the whole group trivial group trivial subgroup 1 1 1 1 1
Total -- -- -- -- -- 146 12

## GAP implementation

### Endomorphisms

The endomorphism structure can be explored using the GAP function Endomorphisms, that requires the SONATA package:

```gap> L := Endomorphisms(SymmetricGroup(5));
[ [ (1,2,3,4,5), (1,2) ] -> [ (), () ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,3)(2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,5)(2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,3)(2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,4)(2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,2)(3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,2)(3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,5)(2,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,4)(2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,2)(4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,4)(3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,3)(4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,5)(3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,4)(3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (2,5)(3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,3)(4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,2,3), (2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,4,5), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,2,3), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,5,4), (1,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,3,2), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,3,2), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,5,4), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,5,2), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,5,3), (3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,4,2), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,4,3), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,2,5), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,4,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,2,5), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,3,4), (1,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,5,2), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,3,4), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,5,2), (3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,4,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,4,2), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,3,2), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,3,5), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,2,4), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,5,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,2,4), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,3,5), (1,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,4,2), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,3,5), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,4,2), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,5,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,5,2), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,5,4), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,3,2), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,3,4), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,5,4), (2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,3,2), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,4,5), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,3,2), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,5,4), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,2,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,2,3), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,3,4), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,3,5), (3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,2,4), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,2,5), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,3,5), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,4,2), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,5,3), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,4,2), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,5,3), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,2,4), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,3,5), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,2,4), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,4,3), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,4,5), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,2,3), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,2,5), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,3,4), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,5,2), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,4,3), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,5,2), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,4,3), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,2,5), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,3,4), (3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,2,5), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,5,3), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,5,4), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,2,3), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,2,4), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,5,2), (1,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,4,2), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,3,2), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,5,2), (1,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,3,2), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,4,2), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,3,5), (2,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,3,4), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,4,3), (2,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,5,3), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,5,4), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,2,3), (1,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,5,3), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,2,3), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,4,3), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,5,3), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,4,3), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,2,5), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,2,4), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,4,2), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,4,5), (2,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,5,2), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,5,4), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,5,2,4), (1,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,5,3,4), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,2,4), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,5,4), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,5,4), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,3,4), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,2,5), (3,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,2,3), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,5,3,2), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,3,5), (2,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,5,2), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,5,3), (2,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,3,4,2,5), (1,3) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,2,4,3,5), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,3,2,5), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (1,2) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,3,2,4,5), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,4,2,3,5), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,2,4), (3,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,2,3), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,4,3,2), (4,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,3,4), (2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (1,5,3,4,2), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,5,2,4,3), (2,5) ] ]
gap> Length(L);
146
gap> M := Filtered(L,x -> x = x*x);
[ [ (1,2,3,4,5), (1,2) ] -> [ (), () ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (4,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (3,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,4) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,2) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (1,3) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (2,5) ],
[ (1,2,3,4,5), (1,2) ] -> [ (), (1,4) ], [ (1,2,3,4,5), (1,2) ] -> [ (), (3,5) ], [ (1,2,3,4,5), (1,2) ] -> [ (1,2,3,4,5), (1,2) ] ]
gap> Length(M);
12
gap> K := List(L,Kernel);
[ Group([ (1,2,3,4,5), (1,2) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]),
Group([ (1,2,3,4,5), (2,4)(3,5) ]), Group([ (1,2,3,4,5), (2,4)(3,5) ]), 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(()), 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(()), 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(()), 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(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()),
Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()), Group(()) ]
gap> FrequencySort(K);
[ [ Group(()), 120 ], [ Group([ (1,2,3,4,5), (1,2) ]), 1 ], [ Group([ (1,2,3,4,5), (2,4)(3,5) ]), 25 ] ]```