A bound for the derived and Frattini subgroups of a prime-power group

Abstract

This paper is based on the seemingly new observation that the Schur multiplier M ( G ) M(G) of a d d -generator group of prime-power order p n p^n has order | M ( G ) | ≤ p d ( 2 n − d − 1 ) / 2 |M(G)|\le p^{d(2n-d-1)/2} . We prove several related results, including sufficient conditions for a sharper bound on | M ( G ) | |M(G)| to be an equality.