Congruence condition on number of subgroups of given order and bounded exponent in class two group for odd prime

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Statement

Suppose p is an odd prime number and P is a group of prime power order for the prime p. Suppose, further, that P is a group of nilpotency class two. Suppose P has a subgroup H of order p^k and exponent p^d.

Then, the number of subgroups of P of order exactly p^k and exponent at most p^d is congruent to 1 mod p.

In other words, the collection of groups of a given order and a given bound on their exponent satisfies a congruence condition within the world of groups of nilpotency class two.

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