Endomorphism structure of symmetric group:S6

This article gives specific information, namely, endomorphism structure, about a particular group, namely: symmetric group:S6.
View endomorphism structure of particular groups | View other specific information about symmetric group:S6

Summary of information

Construct	Value	Order	Second part of GAP ID (if group)
endomorphism monoid	?	1516	--
automorphism group	automorphism group of alternating group:A6	1440	5841
inner automorphism group	symmetric group:S6	720	763
extended automorphism group	direct product of automorphism group and cyclic group:Z2	2880
outer automorphism group	cyclic group:Z2	2

Family contexts

Family name	Parameter values	General discussion of endomorphism structure of family
symmetric group	6	endomorphism structure of symmetric groups

Description of automorphism group

Symmetric group:S6 is the only symmetric group on a finite set in which not every automorphism is inner. For $n\ne 2,6$, the symmetric group $S_n$ is a complete group -- see symmetric groups on finite sets are complete.

$S_6$ is a centerless group, so it is isomorphic to its own inner automorphism group under the conjugation action. The outer automorphism group is cyclic group:Z2, i.e., there is a single non-inner class of outer automorphisms. The automorphism group is thus of order $720\times 2=1440$, with the inner automorphism group as a normal subgroup of order 720 and the outer automorphism group as a quotient group of order 2.

The extension splits, i.e., there exist outer automorphisms that have order 2 as automorphisms.

All outer automorphisms, being in the same equivalence class mod inner automorphisms, induce the same permutation on the set of conjugacy classes. For more on the nature of the permutation, see element structure of symmetric group:S6#Automorphism class structure.

The automorphism group itself is isomorphic to automorphism group of alternating group:A6. In fact, if we identify A6 with PSL(2,9), the automorphism group can be identified with the projective semilinear group of degree two $P\mathrm{\Gamma }L(2,9)$.

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:S6	the whole group	1	1	1440	1440	1
A6 in S6	cyclic group:Z2	three possible conjugacy classes	1	75	1	75	30
the whole group	trivial group	trivial subgroup	1	1	1	1	1
Total	--	--	--	--	--	1516	32

GAP implementation

Automorphisms

We use the functions AutomorphismGroup and InnerAutomorphismsAutomorphismGroup to explore the automorphism structure:

gap> G := SymmetricGroup(6);;
gap> A := AutomorphismGroup(G);
<group with 3 generators>
gap> IdGroup(A);
[ 1440, 5841 ]
gap> B := AutomorphismGroup(AlternatingGroup(6));
<group with 4 generators>
gap> IdGroup(B);
[ 1440, 5841 ]
gap> I := InnerAutomorphismsAutomorphismGroup(A);
<group with 2 generators>
gap> StructureDescription(I);
"S6"

Endomorphisms

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

gap> L := Endomorphisms(SymmetricGroup(6));;
gap> Length(L);
1516
gap> M := Filtered(L, x -> x = x * x);
[ [ (1,2,3,4,5,6), (1,2) ] -> [ (), () ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,6), (1,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (2,6), (2,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,5)(2,3)(4,6), (1,5)(2,3)(4,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (2,3), (2,3) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,3)(2,6)(4,5), (1,3)(2,6)(4,5) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,6)(2,5)(3,4), (1,6)(2,5)(3,4) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,4)(2,3)(5,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,4), (1,4) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (3,4), (3,4) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (4,5), (4,5) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,6)(2,4)(3,5), (1,6)(2,4)(3,5) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,5), (1,5) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,2)(3,4)(5,6), (1,2)(3,4)(5,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (5,6), (5,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,3)(2,4)(5,6), (1,3)(2,4)(5,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (3,5), (3,5) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,5)(2,6)(3,4), (1,5)(2,6)(3,4) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,6)(2,3)(4,5), (1,6)(2,3)(4,5) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,4)(2,6)(3,5), (1,4)(2,6)(3,5) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (4,6), (4,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,2)(3,6)(4,5), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,2)(3,5)(4,6), (1,2)(3,5)(4,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,2), (1,2) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,3)(2,5)(4,6), (1,3)(2,5)(4,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,3), (1,3) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (2,5), (2,5) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (1,4)(2,5)(3,6), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,5)(2,4)(3,6), (1,5)(2,4)(3,6) ],
  [ (1,2,3,4,5,6), (1,2) ] -> [ (3,6), (3,6) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (2,4), (2,4) ], [ (1,2,3,4,5,6), (1,2) ] -> [ (1,2,3,4,5,6), (1,2) ] ]
gap> Length(M);
32
gap> K := List(L,Kernel);;
gap> FrequencySort(K);
[ [ Group(()), 1440 ], [ Group([ (1,2,3,4,5,6), (1,2) ]), 1 ], [ Group([ (1,5,3)(2,6,4), (1,6,5,4,3) ]), 75 ] ]