# Automorph-conjugacy is not finite-join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup)notsatisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it doesnotsatisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about automorph-conjugate subgroup|Get more facts about finite-join-closed subgroup propertyGet more facts about join-closed subgroup property|

## Contents

## Statement

A join of finitely many automorph-conjugate subgroups need not be automorph-conjugate.

## Facts used

### For the Hall subgroups example

- Sylow implies automorph-conjugate
- Hall implies join of Sylow subgroups
- Hall not implies automorph-conjugate

## Proof

### The Hall subgroups example

The proof follows directly from facts (1)-(3).