# Difference between revisions of "Subgroup structure of symmetric group:S4"

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| [[Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:Z4]] || 1 || 3 || -- || -- | | [[Z4 in S4]] || <math>\langle (1,2,3,4) \rangle</math> || [[cyclic group:Z4]] || 1 || 3 || -- || -- | ||

|- | |- | ||

− | | [[normal Klein four-subgroup of S4]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \ | + | | [[normal Klein four-subgroup of S4]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[Klein four-group]] || 1 || 1 || [[symmetric group:S3]] || 1 |

|- | |- | ||

| [[non-normal Klein four-subgroups of S4]] || <math>\langle (1,2), (3,4) \rangle</math> || [[Klein four-group]] || 1 || 3 || -- || -- | | [[non-normal Klein four-subgroups of S4]] || <math>\langle (1,2), (3,4) \rangle</math> || [[Klein four-group]] || 1 || 3 || -- || -- |

## Revision as of 02:29, 21 June 2011

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

View subgroup structure of particular groups | View other specific information about symmetric group:S4

The symmetric group of degree four has many subgroups.

Note that since is a complete group, every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

- The trivial subgroup. Isomorphic to trivial group.(1)
- S2 in S4: The two-element subgroup generated by a transposition, such as . Isomorphic to cyclic group of order two. (6)
- Subgroup generated by double transposition in S4: The two-element subgroup generated by a double transposition, such as . Isomorphic to cyclic group of order two. (3)
- Non-normal Klein four-subgroups of symmetric group:S4The four-element subgroup generated by two disjoint transpositions, such as . Isomorphic to Klein four-group. (3)
- Normal Klein four-subgroup of symmetric group:S4: The unique four-element subgroup comprising the identity and the three double transpositions. Isomorphic to Klein four-group. (1)
- Z4 in S4: The four-element subgroup spanned by a 4-cycle. Isomorphic to cyclic group of order four.(3)
- D8 in S4: The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to dihedral group of order eight. This is also a 2-Sylow subgroup. (3)
- A3 in S4: The three-element subgroup spanned by a three-cycle. Isomorphic to cyclic group of order three.(4)
- S3 in S4: The six-element subgroup comprising all permutations that fix one element. Isomorphic to symmetric group on three elements. (4)
- A4 in S4: The alternating group: the subgroup of all even permutations. Isomorphic to alternating group:A4.(1)
- The whole group.(1)

## Tables for quick information

### Table classifying subgroups up to automorphisms

Automorphism class of subgroups | Representative | Isomorphism class | Number of conjugacy classes | Size of each conjugacy class | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) |
---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 1 | symmetric group:S4 | 1 | |

S2 in S4 | cyclic group:Z2 | 1 | 6 | -- | -- | |

subgroup generated by double transposition in S4 | cyclic group:Z2 | 1 | 3 | -- | 2 | |

A3 in S4 | cyclic group:Z3 | 1 | 4 | -- | -- | |

Z4 in S4 | cyclic group:Z4 | 1 | 3 | -- | -- | |

normal Klein four-subgroup of S4 | Klein four-group | 1 | 1 | symmetric group:S3 | 1 | |

non-normal Klein four-subgroups of S4 | Klein four-group | 1 | 3 | -- | -- | |

S3 in S4 | symmetric group:S3 | 1 | 4 | -- | -- | |

D8 in S4 | dihedral group:D8 | 1 | 3 | -- | -- | |

A4 in S4 | alternating group:A4 | 1 | 1 | cyclic group:Z2 | 1 | |

whole group | symmetric group:S4 | 1 | 1 | trivial group | 0 |

### Table classifying isomorphism types of subgroups

Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|

Trivial group | 1 | 1 | 1 | 1 | 1 | 1 |

Cyclic group:Z2 | 2 | 1 | 9 | 2 | 0 | 0 |

Cyclic group:Z3 | 3 | 1 | 4 | 1 | 0 | 0 |

Cyclic group:Z4 | 4 | 1 | 3 | 1 | 0 | 0 |

Klein four-group | 4 | 2 | 4 | 2 | 1 | 1 |

Symmetric group:S3 | 6 | 1 | 4 | 1 | 0 | 0 |

Dihedral group:D8 | 8 | 3 | 3 | 1 | 0 | 0 |

Alternating group:A4 | 12 | 3 | 1 | 1 | 1 | 1 |

Symmetric group:S4 | 24 | 12 | 1 | 1 | 1 | 1 |

Total | -- | -- | 30 | 11 | 4 | 4 |

### Table listing number of subgroups by order

Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order is congruent to modulo .

Group order | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

1 | 1 | 1 | 1 | 1 |

2 | 9 | 2 | 0 | 0 |

3 | 4 | 1 | 0 | 0 |

4 | 7 | 3 | 1 | 1 |

6 | 4 | 1 | 0 | 0 |

8 | 3 | 1 | 0 | 0 |

12 | 1 | 1 | 1 | 1 |

24 | 1 | 1 | 1 | 1 |

Total | 30 | 11 | 4 | 4 |

### Table listing numbers of subgroups by group property

Group property | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|

Cyclic group | 17 | 5 | 1 | 1 |

Abelian group | 21 | 7 | 2 | 2 |

Nilpotent group | 24 | 8 | 2 | 2 |

Solvable group | 30 | 11 | 4 | 4 |

### Table listing numbers of subgroups by subgroup property

Subgroup property | Occurences as subgroup | Conjugacy classes of occurrences as subgroup | Automorphism classes of occurrences as subgroup |
---|---|---|---|

Subgroup | 30 | 11 | 11 |

Normal subgroup | 4 | 4 | 4 |

Characteristic subgroup | 4 | 4 | 4 |

## Subgroup structure viewed as symmetric group

### Classification based on partition given by orbit sizes

For any subgroup of , the natural action on induces a partition of the set into orbits, which in turn induces an unordered integer partition of the number 4. Below, we classify this information for the subgroups.

Conjugacy class of subgroups | Size of conjugacy class | Induced partition of 4 | Direct product of transitive subgroups on each orbit? | Illustration with representative |
---|---|---|---|---|

trivial subgroup | 1 | 1 + 1 + 1 + 1 | Yes | The subgroup fixes each point, so the orbits are singleton subsets. |

S2 in S4 | 6 | 2 + 1 + 1 | Yes | has orbits |

subgroup generated by double transposition in S4 | 3 | 2 + 2 | No | has orbits |

A3 in S4 | 4 | 3 + 1 | Yes | has orbits |

Z4 in S4 | 3 | 4 | Yes | The action is a transitive group action, so only one orbit. |

normal Klein four-subgroup of S4 | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |

non-normal Klein four-subgroups of S4 | 3 | 2 + 2 | Yes | has orbits |

S3 in S4 | 4 | 3 + 1 | Yes | has orbits |

D8 in S4 | 3 | 4 | Yes | The action is a transitive group action, so only one orbit. |

A4 in S4 | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |

whole group | 1 | 4 | Yes | The action is a transitive group action, so only one orbit. |