# Join-transitively automorph-conjugate subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]

## 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

## Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

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