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This paper is based on the seemingly new observation that the Schur multiplier M(G) of a d-generator group of prime-power order pl has order IM(G) I 1, but only if G/Z is elementary abelian. As a corollary, Wiegold re-derived Green's bound [7] on the order of the Schur multiplier of a prime-power group. Gaschiitz, Neubuiser and Yen [6] subsequently refined Wiegold's methods to obtain a sharper bound on [G, G] (involving the number of generators of the quotient of G by its second centre), and a corresponding sharpening of Green's bound. By a different fine tuning of Wiegold's methods, Jones ([9], [10], [11]) obtained complementary improvements in Green's bound. In this article we reduce the above bounds on [G, G] by incorporating, in turn: (1) the number of generators of G/Z; (2) the abelian group structure of the quotients of the lower central series of G/Z; (3) the restricted Lie algebra structure of the quotients of the lower p-central series of G/Z; (4) the "breadth in G" of the preimages of the generators of G/Z. The fourth approach (which is the only one to involve more than the structure of G/Z) complements a result of Vaughan-Lee [16]. We obtain corresponding reductions in the above-mentioned bounds on the Schur multiplier. As an application we obtain a rough upper bound on the number of d-generator groups of order pn. Furthermore, our results are presented in such a way as to yield bounds on the aEattini subgroup of a prime-power group. Let p be any prime, and let q > 0 be any nonnegative integer multiple of p. (We are interested mainly in the cases q = 0 and q = p.) Given a group G we let Zq (G) denote the subgroup of the centre of G consisting of those elements with order dividing q. Given a normal subgroup N in G, we let N#qG denote the subgroup of G generated by the commutators ngn-1 g-1 and powers nq for n E N, g E G. Received by the editors January 27, 1997. 1991 Mathematics Subject Classification. Primary 20J05. (? 1998 American Mathematical Society

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