Endomorphism structure of symmetric group:S5
From Groupprops
This article gives specific information, namely, element structure, about a particular group, namely: 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 
, 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 ] ]
