Congruence condition fails for subgroups of order p^p and exponent p
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 .
- 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
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 .