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Three-Dimensional Presentations for the Groups of Order at Most 30

Published online by Cambridge University Press:  01 February 2010

Graham Ellis  and
Irina Kholodna
Graham Ellis
Affiliation:
Max-Planck-Institut für Mathematik, D-53225 Bonn, graham.ellis@nuigalway.iehttp://hamilton.nuigalway.ie Mathematics Department, National University of Ireland, Galway, irina.kholodna@nuigalway.ie
Irina Kholodna
Affiliation:
Mathematics Department, National University of Ireland, Galway, irina.kholodna@nuigalway.ie

Abstract

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For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π1X≌ G and π2X = 0, and we quantify the ‘efficiency’ of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π2X2 of the 2-skeleton of X. This module structure can in principle be used to determine the co-homology groups H2(G, A) and H3(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 2 , 1999 , pp. 93 - 117
Copyright
Copyright © London Mathematical Society 1999

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