# Central factor is not finite-join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor)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 (?), .

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## Contents

## Statement

### Statement with symbols

It is possible to have a finite group with subgroups such that both and are central factors of but the product (which in this case is also the join of subgroups ) is not a central factor.

## Definitions used

### Central factor

`Further information: Central factor`

A subgroup of a group is termed a **central factor** if , or equivalently, every inner automorphism of restricts to an inner automorphism of .

## Proof

### Example of the extraspecial group

For any prime , either of the two isomorphism classes of extraspecial groups of order gives a counterexample.