# A3 in S4

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) cyclic group:Z3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

This article is about the subgroup in the group , where is the symmetric group of degree four, acting on the set , and is the three-element subgroup:

In other words, is the alternating group on viewed naturally as a subgroup of .

There are three other conjugate subgroups of in (so a total of four), with each conjugate characterized as the alternating group on some subset of size three. Indexing these by the fixed point, we get:

## Contents

## Complements

The permutable complements to (and also to each of its conjugates) are precisely the 2-Sylow subgroups of , which are the D8 in S4s, namely, copies of dihedral group:D8 sitting inside the whole group. One such complement is:

There are also other lattice complements that are not permutable complements -- for instance, any Z4 in S4 is a lattice complement that is not a permutable complement.

## Arithmetic functions

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

order of whole group | 24 | ||

order of the subgroup | 3 | ||

index of the subgroup | 8 | ||

size of conjugacy class | 4 | ||

number of conjugacy classes in automorphism class | 1 |

## Effect of subgroup operators

Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | S3 in S4 | symmetric group:S3 | |

centralizer | the subgroup itself | (current page) | cyclic group:Z3 |

normal core | trivial subgroup | -- | trivial group |

normal closure | A4 in S4 | alternating group:A4 | |

characteristic core | trivial subgroup | -- | trivial group |

characteristic closure | A4 in S4 | alternating group:A4 |

## Subgroup properties

The subgroup is a Sylow subgroup for the prime 3. Many properties follow from this fact.