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

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This article is about a congruence condition.
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Statement

Suppose p is any prime number and 0 \le k \le 5. Suppose P is a group of prime power order where the prime is p. Then, the number of cyclic subgroups of P of order p^k is congruent to either 0 or 1 modulo p.

Note that the statement is trivially true for p = 2, so it suffices to prove it for odd p.

Facts used

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

Proof

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