Endomorphism structure of symmetric group:S4

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S4.
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 ] ]