# Congruence condition fails for subgroups of order p^p and exponent p

## Statement

Let be a prime number. Suppose is a group of prime power order for the prime . Let be a prime power less than or equal to the order of and be a power of with . It is *not* necessary that the number of subgroups of order and exponent dividing is either zero or congruent to modulo .

## Related facts

### Opposite facts

- Mann's replacement theorem for subgroups of prime exponent
- Congruence condition on number of subgroups of given prime power order
- Congruence condition on Sylow numbers
- Congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group

## Proof

### The case

Consider the dihedral group of order eight. The number of subgroups of order and exponent is equal to , which is neither equal to zero or congruent to modulo .

### The case of odd

Let be an odd prime, and let be the wreath product of groups of order p. Then, is a group of order and exponent . Consider the subgroups of having order and exponent . There are exactly two of these, both of them isomorph-free: the elementary abelian normal subgroup of order , and the semidirect product of the commutator subgroup (which has order ) with the wreathing element of order .