# Klein four-subgroup of alternating group:A5

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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) Klein four-group 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:

This subgroup is isomorphic to the Klein four-group. There are five conjugates (including the subgroup itself) depending on which of the five points is *fixed*:

Fixed point | Subgroup name here | Elements of subgroup |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 |

## Contents

## Arithmetic functions

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

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

order of the subgroup | 4 | |

index of the subgroup | 15 | |

size of conjugacy class of subgroups (equal to index of normalizer) | 3 | |

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 | A4 in A5 | alternating group:A4 | |

centralizer | the subgroup itself | current page | Klein four-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 subgroup 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. | The order of this subgroup is 4, which is the largest power of 2 dividing the order of the group. |

## Related subgroups

### Intermediate subgroups

Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group | Subgroup in intermediate subgroup | Intermediate subgroup in whole group |
---|---|---|---|---|

alternating group:A4 | 1 | V4 in A4 | A4 in A5 |

### Smaller subgroups

Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Number of conjugacy classes of smaller subgroup fixing subgroup and whole group | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|---|

cyclic group:Z2 | 1 | Z2 in V4 | subgroup generated by double transposition in A5 |

## Subgroup properties

### Sylow and corollaries

The subgroup is a 2-Sylow subgroup, so many properties follow as a corollary of that.

Property | Meaning | Why being Sylow implies the property |
---|---|---|

order-conjugate subgroup | conjugate to any subgroup of the same order | Sylow implies order-conjugate |

isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it | (via order-conjugate) |

automorph-conjugate subgroup | conjugate to any subgroup automorphic to it | (via isomorph-conjugate) |

intermediately isomorph-conjugate subgroup | conjugate to any subgroup isomorphic to it inside any intermediate subgroup | Sylow implies intermediately isomorph-conjugate |

intermediately automorph-conjugate subgroup | automorph-conjugate in every intermediate subgroup | (via intermediately isomorph-conjugate) |

pronormal subgroup | conjugate to any conjugate subgroup in their join | Sylow implies pronormal |

weakly pronormal subgroup | (via pronormal) | |

paranormal subgroup | (via pronormal) | |

polynormal subgroup | (via pronormal) |

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 | ||

self-normalizing subgroup | equals its normalizer in the whole group | No | normalizer is A4 in A5 | |

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