# Join of automorph-conjugate subgroups

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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## Definition

A subgroup of a group is termed a **join of automorph-conjugate subgroups** or **AC-generated** if it is generated by a collection of automorph-conjugate subgroups, viz it is a join of automorph-conjugate subgroups.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

automorph-conjugate subgroup | conjugate to all its automorphic subgroups | (obvious) | automorph-conjugacy is not finite-join-closed | |FULL LIST, MORE INFO |

join of Sylow subgroups | join of Sylow subgroups | (obvious) | Any proper nontrivial characteristic subgroup of a -group, e.g., Z2 in Z4 | |FULL LIST, MORE INFO |

Hall subgroup | subgroup whose order and index are relatively prime | (via join of Sylow subgroups) | (via join of Sylow subgroups) | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

closure-characteristic subgroup | its normal closure is a characteristic subgroup | |FULL LIST, MORE INFO | ||

normal-to-characteristic subgroup | if normal, it is also characteristic | |FULL LIST, MORE INFO |

### Conjunction with other properties

Any normal subgroup that is a join of automorph-conjugate subgroups is characteristic. *Thus, this subgroup property is normal-to-characteristic*