# Complements to abelian normal subgroup are automorphic

## Statement

Suppose $G$ is a group, $N$ is an Abelian normal subgroup, and $H, K$ are permutable complements to $N$ in $G$. Then, there is an automorphism $\sigma$ of $G$ such that:

• The restriction of $\sigma$ to $N$ is the identity map on $N$.
• $\sigma$ induces an isomorphism from $H$ to $K$.

In particular, $H$ and $K$ are automorphic subgroups.

(Note: There may exist automorphic subgroups to $H$ that are not permutable complements to $N$).