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

From Groupprops

## Statement

Suppose is an *odd* prime number and is a group of prime power order for the prime . Suppose, further, that is a group of nilpotency class two. Suppose has a subgroup of order and exponent .

Then, the number of subgroups of of order *exactly* and exponent *at most* is congruent to 1 mod .

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.