# Characteristic-isomorph-free not implies normal-isomorph-free in finite

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-isomorph-free subgroup) neednotsatisfy the second subgroup property (i.e., normal-isomorph-free subgroup)

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## Statement

We may have a group with a characteristic subgroup such that there are no other characteristic subgroups isomorphic to it, but there are other normal subgroups isomorphic to it.

## Proof

### Example of an Abelian group

`Further information: prime-cube order group:p2timesp, agemo subgroups of a group of prime power order, omega subgroups of a group of prime power order`

Let be any prime, and let . In other words, is a direct product of cyclic groups of order and respectively.

Consider the subgroup : the set of all elements of that can be written as powers. This set is .

- is a characteristic subgroup: It is defined as the set of powers, so it is clearly invariant under conjugation.
- There is no other characteristic subgroup of of order : All the subgroups of order in lie inside . Further, automorphisms of the form permute all the other subgroups.
- There are other normal subgroups of of order : In fact, there are of them: the other subgroups of .

Thus, is characteristic-isomorph-free in but is *not* normal-isomorph-free in .