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 p-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