# Series-equivalent characteristic subgroups may be distinct

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., series-isomorph-free subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about characteristic subgroup|Get more facts about series-isomorph-free subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not series-isomorph-free subgroup|View examples of subgroups satisfying property characteristic subgroup and series-isomorph-free subgroup

## Statement

It is possible to have a group , a characteristic subgroups and of such that and are isomorphic groups and the quotient groups and are isomorphic groups. In other words, there can be distinct characteristic subgroups that are Series-equivalent subgroups (?).

In particular, a characteristic subgroup of need not be a series-isomorph-free subgroup of .

## Related facts

### Stronger facts

There are two slightly stronger facts that are true, either of which can be used to supply examples:

- Series-equivalent characteristic central subgroups may be distinct
- Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order

## Proof

See the examples in the proof of either of the stronger facts.