# AEP does not satisfy intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., AEP-subgroup)notsatisfying a subgroup metaproperty (i.e., intermediate subgroup condition).

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## Contents

## Statement

### Property-theoretic statement

The subgroup property of being an AEP-subgroup does not satisfy the subgroup metaproperty of the intermediate subgroup condition.

### Statement with symbols

It is possible to have groups such that is an AEP-subgroup of but is not an AEP-subgroup of .

## Proof

### Example of an Abelian group

Let and be isomorphic copies of . Let and be subgroups of order two in and respectively. Then, define:

.

We claim that:

- : This is clear from the definition.
- is an AEP-subgroup of
- is not an AEP-subgroup of : Consider the automorphism of that exchanges the generators of and . This cannot extend to an automorphism of , because in , the generator of is the double of an element, while the generator of is not the double of anything.