# A3 in A4

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) alternating group:A3 and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).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 describes the subgroup in the group . Here, is the alternating group:A4, acting on the set . is the subgroup:

It has three other conjugates:

With this notation, each is the stabilizer of in .

See also subgroup structure of alternating group:A4.

## Contents

## Cosets

Each of the four subgroups has four left cosets and four right cosets. Further, for every pair of subgroups, there is exactly one coset that is a left coset for the first subgroup and a right coset for the second subgroup.

## Complements

All four subgroups have a unique common normal complement, which is the Klein four-subgroup of alternating group:A4:

This is also the unique permutable complement to each of them.

Also, each of has *each* of the following three subgroups as a lattice complement that is not a permutable complement:

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

permutably complemented subgroup | has a permutable complement | Yes | See above | |

lattice-complemented subgroup | has a lattice complement | Yes | See above | |

retract | has a normal complement | Yes | See above | |

complemented normal subgroup | normal subgroup with permutable complement | No | Not normal |

## Arithmetic functions

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

order of whole group | 12 | |

order of subgroup | 3 | |

index | 4 | |

size of conjugacy class | 4 | |

number of conjugacy classes in automorphism class | 1 |

## Effect of subgroup operators

In the table below, we provide values specific to .

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

normalizer | the subgroup itself | current page | cyclic group:Z3 |

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

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

normal closure | whole group | -- | alternating group:A4 |

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

characteristic closure | whole group | -- | alternating group:A4 |

commutator with whole group | subgroup | V4 in S4 | Klein four-group |

## Related subgroups

### Intermediate subgroups

The subgroup is a maximal subgroup of , so there are no strictly intermediate subgroups between and .

### Smaller subgroups

The subgroup is a group of prime order, so it has no proper nontrivial subgroup.

For ease of reference, we take here the subgroup , though the conclusions apply for the other three conjugates as well.

Some of the properties follow on account of being a Sylow subgroup.

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

normal subgroup | equals its conjugate subgroups | No | ||

subnormal subgroup | has a chain to whole group each normal in next | No | ||

pronormal subgroup | Yes | |||

automorph-conjugate subgroup | all automorphic subgroups are conjugate subgroups | Yes | ||

isomorph-conjugate subgroup | Yes | |||

order-conjugate subgroup | Yes | |||

order-automorphic subgroup | Yes | |||

order-isomorphic subgroup | Yes |