# Characteristic not implies quasiautomorphism-invariant

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., quasiautomorphism-invariant subgroup)

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## Statement

It is possible to have a group and a characteristic subgroup of such that is not a quasiautomorphism-invariant subgroup, i.e., there are quasiautomorphisms of that do not send to itself.

## Related facts

- Commutator subgroup not is quasiautomorphism-invariant
- Center is quasiautomorphism-invariant
- Center not is 1-automorphism-invariant
- Quasiautomorphism-invariant not implies 1-automorphism-invariant

## Proof

`Further information: inner automorphism group of wreath product of groups of order p`

Suppose is a prime number greater than . Let be the group isomorphic to the inner automorphism group of the wreath product of two groups of order . is a group of order with an elementary abelian normal subgroup of order , an element of order acting on it from outside, and every non-identity element of has order .

Consider the subgroup . is a group of order , contained inside the elementary abelian normal subgroup. It is a characteristic subgroup.

However, we can construct a quasiautomorphism of that does not preserve as follows: the restriction of to is an automorphism of that fixes the center (which is cyclic of order ) but does not send to itself, and fixes every element of outside . Note that we need to ensure that is strictly bigger than , which is necessary to be able to construct a with the desired specifications.