Join of automorph-conjugate subgroups

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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