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 $N$).

Related facts

• Complement to normal subgroup is isomorphic to quotient
• Hall retract implies order-conjugate: This is the conjugacy part of the Schur-Zassenhaus theorem, and states that, for a normal Hall subgroup, any two complements are not just automorphic, they are also conjugate subgroups.
• Complements to normal subgroup need not be automorphic

Breakdown when the normality constraint is removed or shifted

• Every group of given order is a permutable complement for symmetric groups: A complete breakdown of the analogous statement when the subgroup is no longer assumed to be normal.
• Retract not implies every permutable complement is normal
• Retract not implies normal complements are isomorphic

Other forms of breakdown

• Semidirect product is not left-cancellative for finite groups