# Characteristic not implies fully invariant in class three maximal class p-group

## Contents

## Statement

Let be any prime number. There exists a -group that is of class three that is a Maximal class group (?) with a Characteristic subgroup (?) that is not fully invariant.

## Related facts

### Similar facts

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

### Opposite facts

## Proof

### For odd primes

Suppose is an odd prime. Let be the wreath product of groups of order p and be . is thus a semidirect product of an elementary abelian group of order and a cyclic group of order . Note that for , has exponent but for , has exponent .

Consider , the centralizer of commutator subgroup. is a characteristic subgroup. Consider now an element of order outside (such an element exists because of the way the group is defined as a semidirect product) and a subgroup of of index other than , that does not contain . Consider the retraction from to with kernel . This retraction doesn't preserve , so is not fully invariant.

### For the prime two

`Further information: dihedral group:D16`

Consider the dihedral group:

.

Let . Then, , so is characteristic in . Consider now the subgroup and the element . Consider the retraction to the subgroup with kernel . is not preserved under this retraction (for instance, gets mapped to ), so it is not fully invariant.