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On the Computation of Certain Homotopical Functors

Published online by Cambridge University Press:  01 February 2010

Graham Ellis
Graham Ellis
Affiliation:
Department of Mathematics, National University of IrelandGalway, Ireland, [email protected]

Abstract

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This paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 1 , 1998 , pp. 25 - 41
Copyright
Copyright © London Mathematical Society 1998

References

1. Bacon, M., Kappe, L.C. and Morse, R.F., ‘On the nonabelian tensor square of a 2-Engel group’, Archiv der Mathematik (Basel) 69 (1997) 353–364.CrossRefGoogle Scholar
2. Beyl, F.R. and Tappe, U., Group extensions, representations, and the Schur multiplier, Lecture Notes in Mathematics 958 (Springer, Berlin, 1982).Google Scholar
3. Bosma, W. and Cannon, J.J., Handbook of Magma functions, Third edition (University of Sydney, 1994).Google Scholar
4. Bosma, W., Cannon, J.J and Playoust, C.E., ‘The Magma algebra system I: the user language’, J. Symbolic Computation, 24, 3/4 (1997) 235–265.CrossRefGoogle Scholar
5. Brown, R., Johnson, D.L. and Robertson, E.F., ‘Some computations of nonabelian tensor products of groups’, J. Algebra 111 (1987) 177–202.CrossRefGoogle Scholar
6. Brown, R. and Loday, J.-L., ‘Van Kampen theorems for diagrams of spaces‘, Topology 26 (1987) 311–335.CrossRefGoogle Scholar
7. Burns, J. and Ellis, G., ‘On the nilpotent multipliers of a group’, Mathematische Zeitschrift 226 (1997) 405–428.CrossRefGoogle Scholar
8. Burns, J. and Ellis, G., Inequalities for Baer invariants of finite groups, Canadian Mathematical Bulletin, to appear.Google Scholar
9. Cannon, J.J. and Playoust, C.E., ‘Magma: A new computer algebra system’, Euromath Bulletin, 2, 1 (1996) 113–144.Google Scholar
10. Dennis, R.K., ‘In search of new homology functors having a close relationship to K-theory’, preprint, Cornell University, 1976.Google Scholar
11. Ellis, G., ‘Tensor products and q-crossed modules’, J. London Math. Soc. (2) 51 (1995) 243–258.CrossRefGoogle Scholar
12. Ellis, G., ‘Capability, homology, and central series of a pair of groups’, J. Algebra 179 (1996) 31–46.CrossRefGoogle Scholar
13. Ellis, G., ‘The Schur multiplier of a pair of groups‘, Applied Categorical Structures, to appear (1997).Google Scholar
14. Ellis, G. and Leonard, F., ‘Computing Schur multipliers and tensor products of finite groups’, Proc. Royal Irish Academy (95A) 2 (1995) 137–147.Google Scholar
15. Evens, L., ‘Terminal p-groups‘, Illinois J. Math. 12 (1968) 682–699.CrossRefGoogle Scholar
16. Gilbert, N.D. and Higgins, P.J., ‘The nonabelian tensor product of groups and related constructions‘, Glasgow Mathematical J. 31 (1989) 17–29.CrossRefGoogle Scholar
17. Guin, D., ‘Cohomologie non abéliennes des groupes’, J. Pure Applied Algebra 50 (1988) 109–138.CrossRefGoogle Scholar
18. Hall, P., ‘The classification of prime-power groups’, J. Reine Angew. Math. 182 (1940) 130–141.CrossRefGoogle Scholar
19. Hilton, P.J. and Stammbach, U., A course in homological algebra, Graduate Texts in Mathematics 4 (Springer, New York, 1971).CrossRefGoogle Scholar
20. Kappe, L.-C., ‘The nonabelian tensor product of groups‘, Groups in St. Andrews/Bath 1997, LMS Lecture Notes, (ed.Campbell, C., Robertson, E.F. et al. , Cambridge University Press, Cambridge, 1997).Google Scholar
21. Karpilovsky, G., The Schur multiplier, LMS monographs New Series 2, (Oxford University Press, 1987).Google Scholar
22. Leedham-Green, C.R. and McKay, S., ‘Baer invariants, isologism, varietal laws and homology’, Acta Mathematica 137 (1976) 99–150.CrossRefGoogle Scholar
23. Loday, J.-L., ‘Cohomologie et groupe de Steinberg relatif’, J. Algebra 54 (1978) 178–202.CrossRefGoogle Scholar
24. Lue, A.S.-T., ‘The Ganea map for nilpotent groups‘, J. London Math. Soc. (2) 14 (1976) 309–312.CrossRefGoogle Scholar
25. McDermott, A., ‘Tensor products of groups and related theory’, PhD Thesis, National University of Ireland Galway, May 1998.Google Scholar
26. Miller, C., ‘The second homology group of a group’, Proc. Amer. Math. Soc. 3 (1952) 588–595.CrossRefGoogle Scholar
Supplementary material: File

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