# Number of nilpotent groups equals product of number of groups of order each maximal prime power divisor

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## Statement

Suppose is a natural number with prime factorization:

where are distinct primes and are natural numbers.

Then the number of nilpotent groups of order is the product of the number of groups of order for all :

(Number of groups of order ) (Number of groups of order ) (Number of groups of order )

## Facts used

## Proof

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