# A4 in A5

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) alternating group:A4 and the group is (up to isomorphism) alternating group:A5 (see subgroup structure of alternating group:A5).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

Let be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set . has order .

Consider the subgroup:

is the alternating group on the set fixing the point 5.

has five conjugate subgroups (including the subgroup itself) in , based on the choice of fixed point:

Fixed point | Subgroup name here | Subgroup |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 |

## Contents

## Arithmetic functions

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

order of the whole group | 60 | . See alternating group:A5. |

order of the subgroup | 12 | |

index of the subgroup | 5 | |

size of conjugacy class of subgroup | 5 | |

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 | alternating group:A4 |

centralizer | trivial subgroup | -- | trivial group |

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

normal closure | the whole group | -- | alternating group:A5 |

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

characteristic closure | the whole group | -- | alternating group:A5 |

## Conjugacy class-defining functions

Conjugacy class-defining function | What it means in general | Why it takes this value |
---|---|---|

Sylow normalizer for the prime | A -Sylow subgroup is a subgroup whose order is a power of and index is relatively prime to . Sylow subgroups exist and Sylow implies order-conjugate, i.e., all -Sylow subgroups are conjugate to each other, hence all normalizers of -Sylow subgroups are also conjugate. | The 2-Sylow subgroup is V4 in A5, and its normalizer is precisely this. |

## Related subgroups

### Intermediate subgroups

There are no intermediate subgroups, since the subgroup is a maximal subgroup.

### Smaller subgroups

Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|

Klein-four group | V4 in A4 | V4 in A5 | |

, , , | cyclic group:Z3 | A3 in A4 | A3 in A5 |

, , | cyclic group:Z2 | subgroup generated by double transposition in A4 | subgroup generated by double transposition in A5 |

## Description in alternative interpretations of the whole group

Description of | Corresponding description of |
---|---|

special linear group of degree two over field:F4, i.e., | Borel subgroup, i.e., subgroup of upper-triangular invertible matrices where the two diagonal entries are mutual inverses. See more at Borel subgroup of special linear group of degree two. |

projective special linear group of degree two over field:F5, i.e., | Unclear/nothing concise (?) |

## Subgroup properties

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

normal subgroup | equals all its conjugate subgroups | No | (see above for other conjugate subgroups) | |

2-subnormal subgroup | normal subgroup in its normal closure | No | ||

subnormal subgroup | series from subgroup to whole group, each normal in next | No | ||

contranormal subgroup | normal closure is whole group | Yes | ||

abnormal subgroup | For any element , we have | Yes | Follows from being a Sylow normalizer, see Sylow normalizer implies abnormal | |

weakly abnormal subgroup | Yes | Follows from being abnormal | ||

self-normalizing subgroup | equals normalizer in the whole group | Yes | ||

self-centralizing subgroup | contains its centralizer in the whole group | Yes | ||

subgroup whose join with any distinct conjugate is the whole group | join of the subgroup with any distinct conjugate subgroup is the whole group | Yes | ||

maximal subgroup | no proper subgroup containing it | Yes | Note that in a finite solvable group, any maximal subgroup has prime power index (see maximal subgroup has prime power index in finite solvable group). The fact that we have a maximal subgroup here whose index is not a prime power is consistent with the fact that alternating group:A5 is not solvable. | |

pronormal subgroup | any conjugate to it is conjugate in their join | Yes | ||

weakly pronormal subgroup | Yes | |||

paranormal subgroup | Yes | |||

polynormal subgroup | Yes |

### Other properties

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

Hall subgroup | order and index are relatively prime | Yes | -Hall subgroup. Also, order (12) and index (5) are relatively prime. | |

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