# Join-transitively automorph-conjugate subgroup

From Groupprops

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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]

## 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 of a group is termed **join-transitively automorph-conjugate** if, for any automorph-conjugate subgroup of , the subgroup 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

## 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 are groups such that both and are automorph-conjugate in . Then, is also automorph-conjugate in . |