Upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order

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Statement

Suppose p is a prime number and G is a finite p-group of order p^n. Then, the Schur multiplier of G is also a finite p-group and its order is at most p^{n(n-1)/2}. In fact, we can say something stronger:

\! |G'||M(G)| \le p^{n(n-1)/2}

where G' denotes the derived subgroup of G.

Related facts

Stronger facts about Schur multiplier