Proficient presentations and direct products of finite groups

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R]; let d_G(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference d_G(R/[R, R]) - d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider d_F(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that d_F(R) = d_G(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time. The first part of the paper discusses similar issues in the category of profinite groups and continuous homomorphisms. One of the conclusions is that a finite group is proficient as discrete group if and only if it is efficient as profinite group. Returning to the discrete setting, the second part explores the proficiency of a direct product in terms of conditions on the direct factors.