Congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group

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This article is about a congruence condition.
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Statement

Let p be a prime number and suppose P is an abelian group of prime power order, i.e., a finite p-group. Suppose p^k is a power of p less than or equal to the order of P and p^d is a power of p less than or equal to p^k.

Then, the number of subgroups of P of order p^k and exponent dividing p^d is either equal to zero or congruent to 1 modulo p.

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