# Groups embeddable as normal subgroups in a finite group with a common complement

## Definition

Suppose $G$ and $H$ are finite groups. We say that $G$ and $H$ are embeddable sd normal subgroups in a finite group with a common complement if there exists a finite group $C$ with a normal subgroup $A$ isomorphic to $G$, a normal subgroup $B$ isomorphic to $H$, and a subgroup $D$ such that $A,D$ are permutable complements and $B,D$ are permutable complements.

Equivalently $D$ is a retract of $C$ having both $A$ and $B$ as normal complements.

## Facts

• If we remove the condition of normality on both embeddings, then the only constraint we get is that the two groups have the same order. This follows from the fact that any finite group of order $n$ is a permutable complement to the symmetric group on $n-1$ letters in the symmetric group on $n$ letters by Cayley's theorem. For full proof, refer: every group of given order is a permutable complement for symmetric groups
• If we impose the additional constraint that the permutable complement itself be a normal subgroup, then the two groups are forced to be isomorphic. For full proof, refer: Direct product is cancellative for finite groups