# Congruence condition fails for number of characteristic subgroups in group of prime power order

From Groupprops

## Contents

## Statement

Let be any prime number.

It is possible to have the following situation: is a finite p-group of order , and there exists with such that the number of Characteristic subgroup (?)s (see also Characteristic subgroup of group of prime power order (?)) of of order is a nonzero number that is not congruent to 1 mod .

## Related facts

### Stronger facts

### Opposite facts

- Congruence condition on number of subgroups of given prime power order tells us that the opposite is true if we are looking at
*all*subgroups or at all normal subgroups.

### Similar facts

- Congruence condition fails for number of normal subgroups of given prime power order: Note that this states that the number of normal subgroups of a given prime power order may be a nonzero number not congruent to 1 modulo the prime,
*but to construct a counterexample, we need to move to an ambient finite group that is not itself a p-group*.

## Proof

### Case

`Further information: central product of D8 and Z8`

Consider to be the central product of D8 and Z8, a group of order 32 (GAP ID: (32,38)). has two characteristic subgroups of order 8:

- The center, which is isomorphic to cyclic group:Z8.
- An isomorph-free subgroup isomorphic to the quaternion group.