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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, [email protected]://hamilton.nuigalway.ie Mathematics Department, National University of Ireland, Galway, [email protected]
Irina Kholodna
Affiliation:
Mathematics Department, National University of Ireland, Galway, [email protected]

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

References

1.Baik, Y.-G., Harlander, J. and Pride, S.J., ‘The geometry of group extensions’, J. Group Theory 1 (1998) 395–16.Google Scholar
2.Baues, H. J.,Combinatorial homotopy and 4-dimensional complexes, de Gruyter Expos. Math. vol. 2 (de Gruyter, Berlin, New York, 1991).Google Scholar
3.Beyl, F. R. and Tappe, J., Group extensions, representations, and the Schur multiplier, Lecture Notes in Math. 958 (Springer, 1982).Google Scholar
4.Bogley, W. A. and Pride, S. J., ‘Calculating generators of π2 ’, Two-dimensional homotopy and combinatorial group theory (ed. Hog-Angeloni, C., Metzler, W. and Sieradski, A. J.), London Mathematical Society Lecture Note Series 197 (Cambridge University Press, 1993) 157–188.CrossRefGoogle Scholar
5.Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system I: the user language’, J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
6.Brown, R. and Higggins, P. J., ‘Tensor products and homotopies for ω-groupoids and crossed complexes–, J. Pure Applied Algebra 47 (1987) 1–33.CrossRefGoogle Scholar
7.Brown, R. and Huebschmann, J., ‘Identities among relations’, Low dimensional topology (ed. Brown, R. and Thickstun, T. L.), London Mathematical Society Lecture Note Series 48 (1982) 153–202.CrossRefGoogle Scholar
8.Brown, R., Johnson, D. L. and Robertson, E. F., ‘Some computations of nonabelian tensor products of groups’, J. Algebra 111 (1987) 177–202CrossRefGoogle Scholar
9.Brown, R. and Porter, T., ‘On the Schreier theory of nonabelian extensions: generalisations and computations’, Proc. Royal Irish Academy 96 (1996) 213–227.Google Scholar
10.Brown, R. and Razak Salleh, A., ‘Free crossed resolutions of groups and presentations of modules of identities among relations’, LMS J. Comput. Math. 2 (1999) 28–61 http://www.lms.ac.uk/jcm/2/lms99001.CrossRefGoogle Scholar
11.Chou, T.-W. J. and Collins, G. E., ‘Algorithms for the solution of linear diophantine equations’, SIAM J. Computing 11 (1982) 687–708.CrossRefGoogle Scholar
12.Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups, 4th edn (Springer-Verlag, 1979).Google Scholar
13.Dyer, M., ‘Crossed and precrossed modules’, Two-dimensional homotopy and combinatorial group theory (ed. Hog-Angeloni, C., Metzler, W. and Sieradski, A. J.), London Mathematical Society Lecture Note Series 197 (Cambridge University Press, 1993) 125–156.CrossRefGoogle Scholar
14.Ellis, G., ‘Nonabelian exterior products of groups and an exact sequence in the homology of groups’, Glasgow Math. J. 29 (1987) 13–19.CrossRefGoogle Scholar
15.Ellis, G., ‘Crossed squares and combinatorial homotopy’, Math. Zeitschrift 214 (1993) 93–110.CrossRefGoogle Scholar
16.Ellis, G., ‘On the computation of certain homotopical functors’, LMSJ. Comput.Math. 1 (1998) 25–1 http://www.lms.ac.uk/jcm/l/lms97004.CrossRefGoogle Scholar
17.Ellis, G., ‘Enumerating prime-power homotopy k-types’, Math. Zeitschrift (to appear).Google Scholar
18.Ellis, G. and McDermott, A., ‘Tensor products of prime-power groups’, J. Pure Applied Algebra 132 (1998) 119–128.CrossRefGoogle Scholar
19.Ellis, G. and Porter, T., ‘Free and projective crossed modules and the second homology group of a group’, J. Pure Applied Algebra 40 (1986) 27–31.CrossRefGoogle Scholar
20.Epstein, D.B. A., ‘Finite presentations of groups and 3-manifolds’, Quart. J. Math. Oxford (2) 12 (1961) 205–212.CrossRefGoogle Scholar
21. THE GAP GROUP, GAP - Groups, Algorithms and Programming (School of Mathematical and Computational Sciences, University of St. Andrews, Scotland, 1998).Google Scholar
22.Groves, J. R. J., ‘An algorithm for computing homology groups’, J. Algebra 194 (1997) 331–361.CrossRefGoogle Scholar
23.Harlander, J., ‘Closing the relation gap by direct product stabilization’, J. Algebra 182 (1996) 511–521.CrossRefGoogle Scholar
24.Kovács, L. G., ‘Finite groups with trivial multiplicator and large deficiency’, ‘Groups-Korea ’ 94 (Pusan) (de Gruyter, Berlin, 1995) 211–225.Google Scholar
25.Johansson, L., Lambe, L. and Sköldberg, E., ‘Normal forms and iterative methods for constructing resolutions’, Preprint, Stockholm, 1998.Google Scholar
26.Johnson, D. L., Topics in the theory of group presentations, London Mathematical Society Lecture Note Series 42 (Cambridge University Press, 1980).CrossRefGoogle Scholar
27.Lenstra, A. K., Lenstra, H. W. and Lovasz, L., ‘Factoring polynomials with rational coefficients’, Math. Annalen 261 (1982) 515–534.CrossRefGoogle Scholar
28.Ratcliffe, J. G., ‘Free and projective crossed modules’, J. London Math. Soc. 22 (1980)66–74.CrossRefGoogle Scholar
29.Rotman, J.J., An introduction to algebraic topology, Graduate Texts in Math.(Springer-Verlag, 1988).CrossRefGoogle Scholar
30.Sieradski, A.J., ‘Algebraic topology for two-dimensional complexes’, Two-dimensional homotopy and combinatorial group theory (ed. Angeloni, C. Hog-, Metzler, W.Sieradski, A. J.), London Mathematical Society Lecture Note Series 197 (Cambridge University Press, 1993) 51–96.CrossRefGoogle Scholar
31.Swan, R.G., ‘Minimal resolutions for finite groups’, Topology 4 (1965) 193–208.CrossRefGoogle Scholar
32.Whitehead, J.H.C., ‘Combinatorial homotopy II’, Bull. American Math. Soc. 55 (1949)453–496.CrossRefGoogle Scholar
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