# Composition factor-equivalent not implies embeddable as normal subgroups in a finite group with isomorphic quotient groups

From Groupprops

## Statement

We can have two groups and that are composition factor-equivalent: in other words, they have the same list of composition factors (possibly in a different order) but such that they are not embeddable as normal subgroups in a finite group with isomorphic quotient groups.

## Facts used

- Complete and composition factor-equivalent not implies isomorphic: There can exist finite complete groups and that are composition factor-equivalent but not isomorphic.
- Equivalence of definitions of complete group: A group is complete if and only if whenever it is embedded as a normal subgroup in another group, it is actually a direct factor of that group.
- Direct product is cancellative for finite groups: If for finite groups , then .

## Proof

### Construction

By Fact (1), we have that there exist finite non-isomorphic complete groups and such that and are composition factor-equivalent. We prove that these and work for our purpose, i.e., we will show that they *cannot* be embedded as normal subgroups in the same group with isomorphic quotient groups.

### Proof by contradiction

**Assumption from which we derive the contradiction**: Suppose there exists a group with a normal subgroup isomorphic to and a normal subgroup isomorphic to , such that .

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Expanation |
---|---|---|---|---|---|

1 | There exist subgroups (possibly equal, possibly distinct) such that is an internal direct product of and (and hence is isomorphic to the external direct product ) and is also an internal direct product of and (and hence is isomorphic to the external direct product ). | Fact (2) | |||

2 | and . | Step (1) | Follows from basic properties of internal direct products. | ||

3 | . | Step (2) | Step-given direct. | ||

4 | is isomorphic both to and . Thus, . | Steps (1), (3) | Use that to replace by in the expression . | ||

5 | . | Fact (3) | is finite. | Step (4) | direct. |

6 | , the desired contradiction. | We use that . | Step (5) | direct. |