Endomorphism structure of 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.

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.

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 ] ]