All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order

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This article gives a fact that is true for small groups of prime power order.More specifically, it is true for all groups of order p^k where k is at most equal to 5.
See more such facts| See more facts true for prime powers up to the same maximum power 5

Statement

Suppose p is a prime number and k is an integer satisfying 0 \le k \le 5. Suppose P is a group of order p^k. Then, P is a finite group in which all cumulative conjugacy class size statistics values divide the order of the group.

In this case, it means that for any r \le k, the number of elements of P whose conjugacy class has size dividing p^r is itself a power of p.

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