caa a22 4500 388091673 CHVBK 20180307125253.0 cr unu---uuuuu 161130e199910 xx s 000 0 eng 10.1017/S0004972700036315 doi S0004972700036315 pii (NATIONALLICENCE)cambridge-10.1017/S0004972700036315 Proficient presentations and direct products of finite groups [Elektronische Daten] 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 dg(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 dG(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 dF(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 dF(R) = dG(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. Copyright © Australian Mathematical Society 1999 Gruenberg K.W. Queen Mary and Westfield College, Mile End Road, London El 4NS, England, e-mail: k.w.gruenberg@qmw.ac.uk Kovács L.G. School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia, e-mail: kovacs@maths.anu.edu.au Bulletin of the Australian Mathematical Society Cambridge University Press 60/2(1999-10), 177-189 0004-9727 60:2<177 1999 60 BAZ https://doi.org/10.1017/S0004972700036315 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0004972700036315 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gruenberg K.W. Queen Mary and Westfield College, Mile End Road, London El 4NS, England, e-mail: k.w.gruenberg@qmw.ac.uk NATIONALLICENCE 700 1- Kovács L.G. School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia, e-mail: kovacs@maths.anu.edu.au NATIONALLICENCE 773 0- Bulletin of the Australian Mathematical Society Cambridge University Press 60/2(1999-10), 177-189 0004-9727 60:2<177 1999 60 BAZ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge