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

From Groupprops

## Definition

Suppose and are finite groups. We say that and are **embeddable sd normal subgroups in a finite group with a common complement** if there exists a finite group with a normal subgroup isomorphic to , a normal subgroup isomorphic to , and a subgroup such that are permutable complements and are permutable complements.

Equivalently is a retract of having both and as normal complements.

## Relation with other relations

### Weaker relations

- Groups embeddable as normal subgroups in a finite group with isomorphic complements
- Groups embeddable as normal subgroups in a finite group with isomorphic quotient groups
- Composition factor-equivalent groups:
`For full proof, refer: Embeddable as normal subgroups in a finite group with a common complement implies composition factor-equivalent`

## 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 is a permutable complement to the symmetric group on letters in the symmetric group on 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`