# Congruence condition fails for number of normal subgroups of given prime power order

## Contents

## Statement

It is possible to have a finite group , and a prime power dividing the order of , such that the number of normal subgroups of of order is a nonzero number that is *not* congruent to 1 mod .

Note that any example must have as *not* being a finite -group itself, because of the congruence condition on number of subgroups of given prime power order and the equivalence of definitions of universal congruence condition.

## Related facts

### Opposite facts

## Proof

`Further information: direct product of A4 and Z4, direct product of A4 and V4`

### Example of direct product of A4 and Z4

Let be the group direct product of A4 and Z4. This is a group of order 48 (GAP ID: (48,31)) obtained as the external direct product of alternating group:A4 (order 12) and cyclic group:Z4 (order 4). Suppose we denote:

where is alternating group:A4 and is cyclic group:Z4.

Note that the order is of the form:

The group has exactly two normal subgroups of order 4:

- The subgroup where is the subgroup in corresponding to V4 in A4, i.e., the subgroup . This subgroup is isomorphic to a Klein four-group.
- The subgroup . This subgroup is isomorphic to cyclic group:Z4.

Thus, the number of subgroups of order is 2, which is a *nonzero* number that is not congruent to 1 mod 2.

### Example of direct product of A4 and V4

Let be the group direct product of A4 and Z4. This is a group of order 48 (GAP ID: (48,31)) obtained as the external direct product of alternating group:A4 (order 12) and Klein four-group (order 4). Suppose we denote:

where is alternating group:A4 and is Klein four-group.

Note that the order is of the form:

The group has exactly two normal subgroups of order 4:

- The subgroup where is the subgroup in corresponding to V4 in A4, i.e., the subgroup . This subgroup is isomorphic to a Klein four-group.
- The subgroup . This subgroup is isomorphic to Klein four-group.

Thus, the number of subgroups of order is 2, which is a *nonzero* number that is not congruent to 1 mod 2.