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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

References

[1]. Alperin, J.L., Local representation theory (Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
[2]Campbell, C.M. and Robertson, E.F., ‘Deficiency zero presentation for SL(2, p)’, Bull. London Math. Soc. 12 (1980), 17–20.CrossRefGoogle Scholar
[3]Campbell, C.M., Robertson, E.F. and Williams, P.D., ‘Efficient presentations of direct powers of imperfect groups’, Algebra Colloq. 4 (1997), 21–27.Google Scholar
[4]Cossey, J., Gruenberg, K.W. and Kovács, L.G., ‘The presentation rank of a direct product of finite groups’, J. Algebra 28 (1974), 597–603.Google Scholar
[5]Curtis, C.W. and Reiner, I., Methods of representation theory I (Wiley, New York, 1981).Google Scholar
[6]Dunwoody, M.J., ‘Relation modules’, Bull. London Math. Soc. 4 (1972), 151–155.CrossRefGoogle Scholar
[7]Gruenberg, K.W., ‘Über die Relationenmoduln einer endlichen Gruppe’, Math. Z. 118 (1970), 30–33.CrossRefGoogle Scholar
[8]Gruenberg, Karl W., Relation modules of finite groups, Conference Board of the Mathematical Sciences Regional Conference Series in Math. 25 (Amer. Math. Soc, Providence, R.I., 1976).CrossRefGoogle Scholar
[9]Gruenberg, K.W., ‘The partial Euler characteristics of the direct powers of a finite group’, Arch. Math. 35 (1980), 267–274.Google Scholar
[10]Harlander, J., ‘Closing the relation gap by direct product stabilization’, J. Algebra 182 (1996), 511–521.Google Scholar
[11]Kenne, P.E., Minimal group presentations: a computational approach, PhD thesis (Australian National University, Canberra, ACT, 1991).Google Scholar
[12]Kovács, L.G., ‘Finite groups with trivial multiplicator and large deficiency’, in Groups – Korea‘94, (Kim, A.C. and Johnson, D.L., Editors), Proc. Internat. Conf. Pusan Nat. Univ. August 18–25, 1994 (de Gruyter, Berlin, New York, 1995), pp. 211–225.Google Scholar
[13]Swan, R.G., ‘Minimal resolutions for finite groups’, Topology 4 (1965), 193–208.Google Scholar
[14]Wilson, J.S., Profinite groups, London Math. Soc. Monographs (Clarendon Press, Oxford, 1998).Google Scholar
[15]Wotherspoon, C.I., The deficiency of particular finite groups, Ph.D. Thesis (University of St. Andrews, Scotland, 1995).Google Scholar