# Characteristic not implies characteristic-isomorph-free in finite

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., characteristic-isomorph-free subgroup)

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## Contents

## Statement

We can find a group (in fact, we can choose to be a finite group) and characteristic subgroups of that are not isomorphic to each other but distinct.

## Related facts

- Characteristic-isomorph-free not implies normal-isomorph-free
- Normal-isomorph-free not implies isomorph-free
- Characteristic equals characteristic-isomorph-free in finite abelian group

## Proof

### Example of the infinite cyclic group

Let be the infinite cyclic group: the group of integers under addition. Then, is a characteristic subgroup of for any , and all the are isomorphic for a nonnegative integer. Thus, there are distinct characteristic subgroups that are isomorphic.

### Example of a finite solvable group

`Further information: symmetric group:S3`

Let be a nontrivial metabelian group that is also a centerless group, with derived subgroup . Let be a group isomorphic to . Then, in the direct product , we have:

- The subgroup equals the center of , hence is characteristic.
- The subgroup equals the derived subgroup of , hence is characteristic.

Thus, we have two distinct characteristic subgroups that are isomorphic.

A particular case of this is where is the symmetric group on three letters. In this case, is A3 in S3 and is cyclic of order three.

### Example of a finite nilpotent group

There are groups of order with characteristic maximal subgroups that are isomorphic. For instance, the group with GAP Group ID has three pairwise isomorphic characteristic subgroups (each with group ID ).