# Groups embeddable as normal subgroups in a finite group with isomorphic quotient groups

From Groupprops

## Definition

Suppose and are finite groups. We say that and are **embeddable as normal subgroups in a finite group with isomorphic quotient groups** if there exists a group with normal subgroups such that:

- is isomorphic to .
- is isomorphic to .
- is isomorphic to .

## Relation with other relations

### Stronger relations

- Groups embeddable as normal subgroups in a finite group with a common complement
- Groups embeddable as normal subgroups in a finite group with isomorphic complements

### Weaker relations

- Composition factor-equivalent groups:
*For proof of the implication, refer Embeddable as normal subgroups in a finite group with isomorphic quotient groups implies composition factor-equivalent and for proof of its strictness (i.e. the reverse implication being false) refer Composition factor-equivalent not implies embeddable as normal subgroups in a finite group with isomorphic quotient groups*.