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Proficient presentations and direct products of finite groups

Published online by Cambridge University Press:  17 April 2009

K.W. Gruenberg  and
L.G. Kovács
K.W. Gruenberg
Affiliation:
Queen Mary and Westfield College, Mile End Road, London El 4NS, England, e-mail: k.w.gruenberg@qmw.ac.uk
L.G. Kovács
Affiliation:
School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia, e-mail: kovacs@maths.anu.edu.au
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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 2 , October 1999 , pp. 177 - 189
Copyright
Copyright © Australian Mathematical Society 1999

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