# Characteristic not implies fully invariant in odd-order class two p-group

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a odd-order class two p-group. That is, it states that in a odd-order class two p-group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., fully invariant subgroup)

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## Statement

For any (odd) prime , there exists a -group of class two and a characteristic subgroup of this group that is not fully invariant.

The construction also works for , but for , there are already examples of abelian groups with characteristic subgroups that are not fully invariant.

## Related facts

- Characteristic not implies fully invariant in finite abelian group
- Characteristic equals fully invariant in odd-order abelian group
- Characteristic not implies fully invariant in odd-order class three maximal class group
- Center not is fully invariant in odd-order class two p-group
- Socle not is fully invariant in odd-order class two p-group

## Proof

Let be an odd prime. Let be any non-abelian group of order with center . There are two possibilities for : a group of prime-square exponent, and a group of prime exponent. In both such groups, there is an element of order outside .

Define where is the cyclic group of order with generator . The center of is the subgroup . Then:

- is characteristic in , because center is characteristic.
- is not fully invariant in : Consider the retraction with kernel and with image generated by the element . This is an endomorphism of , but it does not send to itself, since the element gets sent to , which is outside .