Join-transitively automorph-conjugate subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed join-transitively automorph-conjugate if its join with any automorph-conjugate subgroup is an automorph-conjugate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed join-transitively automorph-conjugate if, for any automorph-conjugate subgroup $K$ of $G$, the subgroup $\langle H,K\rangle$ is also an automorph-conjugate subgroup.

Formalisms

In terms of the join-transiter

This property is obtained by applying the join-transiter to the property: automorph-conjugate subgroup
View other properties obtained by applying the join-transiter

Relation with other properties

Stronger properties

• Characteristic subgroup: For proof of the implication, refer Characteristic implies join-transitively automorph-conjugate and for proof of its strictness (i.e. the reverse implication being false) refer Join-transitively automorph-conjugate not implies characteristic.

Weaker properties

• Automorph-conjugate subgroup
• Join of automorph-conjugate subgroups
• Closure-characteristic subgroup
• Core-characteristic subgroup

Incomparable properties

• Intermediately automorph-conjugate subgroup
• Intersection-transitively automorph-conjugate subgroup

Metaproperties

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Here is a summary:

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
finite-join-closed subgroup property	Yes	(obvious)	0	Suppose $H,K\leq G$ are groups such that both $H$ and $K$ are automorph-conjugate in $G$. Then, $\langle H,K\rangle$ is also automorph-conjugate in $G$.