# S4 in S5

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) symmetric group:S4 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Contents

## Definition

The group is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take as the symmetric group on the set .

We take as the subgroup fixing , so is symmetric group:S4 acting on the set .

has four other conjugate subgroups, each corresponding to a different fixed point:

- is the subgroup fixing , and is the symmetric group on the set .
- is the subgroup fixing , and is the symmetric group on the set .
- is the subgroup fixing , and is the symmetric group on the set .
- is the subgroup fixing , and is the symmetric group on the set .
- is the subgroup fixing , and is the symmetric group on the set .

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of group | 120 | |

order of subgroup | 24 | |

index of subgroup | 5 | |

size of conjugacy class of subgroup | 5 | |

number of conjugacy classes in automorphism class of subgroup | 1 | |

size of automorphism class of subgroup | 5 |

## Subgroup properties

### Other properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

Hall subgroup | order and index are relatively prime | Yes | -Hall subgroup. Also, order (24) and index (5) are relatively prime. | |

p-complement | complement of a -Sylow subgroup | Yes | -complement for |

## GAP implementation

The group-subgroup pair can be constructed as follows:

`G := SymmetricGroup(5); H := SymmetricGroup(4);`