# Congruence condition on number of cyclic subgroups of small prime power order

From Groupprops

This article is about a congruence condition.

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## Statement

Suppose is any prime number and . Suppose is a group of prime power order where the prime is . Then, the number of cyclic subgroups of of order is congruent to either or modulo .

Note that the statement is trivially true for , so it suffices to prove it for odd .

## Facts used

- Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
- Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime

## Proof

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