# AEP does not satisfy intermediate subgroup condition

## 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 $H \le K \le G$ such that $H$ is an AEP-subgroup of $G$ but $H$ is not an AEP-subgroup of $K$.

## Proof

### Example of an Abelian group

Let $A$ and $B$ be isomorphic copies of $\mathbb{Z}/4\mathbb{Z}$. Let $C$ and $D$ be subgroups of order two in $A$ and $B$ respectively. Then, define:

$G = A \times B, H = C \times D, K = C \times B$.

We claim that:

• $H \le K \le G$: This is clear from the definition.
• $H$ is an AEP-subgroup of $G$
• $H$ is not an AEP-subgroup of $K$: Consider the automorphism of $H$ that exchanges the generators of $C$ and $D$. This cannot extend to an automorphism of $K$, because in $K$, the generator of $D$ is the double of an element, while the generator of $C$ is not the double of anything.