# Group of prime power order may have multiple characteristic subgroups of prime order

From Groupprops

## Contents

## Statement

Let be a prime number. Then, there exists a group of prime power order that has more than one characteristic subgroup of order .

## Related facts

- Maximal among abelian characteristic subgroups may be multiple and isomorphic
- Characteristic maximal not implies isomorph-free in group of prime power order

## Proof

### Case

`Further information: SmallGroup(16,4), subgroup structure of SmallGroup(16,4)`

For , we can take:

.

Then, has three characteristic subgroups of order two:

- The subgroup is the commutator subgroup.
- The subgroup is the unique subgroup generated by the unique element that is a square but not a commutator.
- The subgroup is the unique subgrou pgenerated by the element that is a producto f squraes but not a square itself.