Endomorphism structure of symmetric group:S4
This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S4.
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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 , 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 | ? | ? | -- |
| 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 |
Description of automorphism group
Symmetric group:S3 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.
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:S4 | the whole group | 1 | 1 | 24 | 24 | 1 |
| normal V4 in S4 | symmetric group:S3 | S3 in S4 (any of four conjugate subgroups) | 1 | 4 | 6 | 24 | 4 |
| 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 | 6 |
| whole group | trivial group | trivial subgroup | 1 | 1 | 1 | 1 | 1 |
| Total | -- | -- | -- | -- | -- | 58 | 12 |