Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T18:55:41.635Z Has data issue: false hasContentIssue false
  • English
  • Français

FINITELY ANNIHILATED GROUPS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 June 2014

MAURICE CHIODO
MAURICE CHIODO*
Affiliation:
Mathematics Department, University of Neuchâtel, Rue Emile-Argand 11, Neuchâtel 2000, Switzerland email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1976, Wiegold asked if every finitely generated perfect group has weight 1. We introduce a new property of groups, finitely annihilated, and show that this might be a possible approach to resolving Wiegold’s problem. For finitely generated groups, we show that in several classes (finite, solvable, free), being finitely annihilated is equivalent to having noncyclic abelianisation. However, we also construct an infinite family of (finitely presented) finitely annihilated groups with cyclic abelianisation. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. This recovers the known partial results on the Wiegold problem: a finite (or finitely generated solvable) perfect group has weight 1.

Keywords

Wiegold’s problemcoverings of groupsweight of finite groupsperfect groups

MSC classification

Secondary: 20F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 404 - 417
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Baumslag, B., ‘Residually free groups’, Proc. Lond. Math. Soc. 17(3) (1967), 402–418.CrossRefGoogle Scholar
Bridson, M. and Wilton, H., ‘The triviality problem for profinite completions’, (2014),arXiv:1401.2273v1.Google Scholar
Brodie, M. A., ‘Finite n-coverings of groups’, Arch. Math. 63 (1994), 385–392.Google Scholar
Brodie, M. A., Chamberlain, R. F. and Kappe, L.-C., ‘Finite coverings by normal subgroups’, Proc. Amer. Math. Soc. 104(3) (1988), 669–674.CrossRefGoogle Scholar
Chiodo, M., ‘Finding non-trivial elements and splittings in groups’, J. Algebra 331 (2011), 271–284.CrossRefGoogle Scholar
de la Harpe, P., Topics in Geometric Group Theory (The University of Chicago Press, Chicago, 2000).Google Scholar
Hempel, J., ‘Residual finiteness for 3-manifolds’, in: Combinatorial Group Theory and Topology, Alta, Utah, 1984, Annals of Mathematics Studies, 111 (Princeton University Press, Princeton, NJ, 1987), 379–396.Google Scholar
Howie, J., ‘A proof of the Scott–Wiegold conjecture on free products of cyclic groups’, J. Pure Appl. Algebra 173(2) (2002), 167–176.Google Scholar
Hungerford, T., Algebra (Springer, New York, 1974).Google Scholar
Klyachko, A. A., ‘A funny property of sphere and equations over groups’, Comm. Algebra 21(7) (1993), 2555–2575.Google Scholar
Kutzko, P., ‘On groups of finite weight’, Proc. Amer. Math. Soc. 55(2) (1976), 279–280.Google Scholar
Mazurov, V. D. and Khukhro, E. I. (eds.) The Kourovka Notebook (Siberian Division of Russian Academy of Sciences, Institute of Mathematics, Novosibirsk, 2006).Google Scholar
Miller, C. F. III, ‘Decision problems for groups—survey and reflections’, in: Algorithms and Classification in Combinatorial Group Theory, Mathematical Sciences Research Institute Publications, 23 (eds. Baumslag, G. and Miller, C. F. III) (Springer, New York, 1992), 1–59.Google Scholar
Perelman, G., ‘Ricci flow with surgery on three-manifolds’, (2003), arXiv:math/0303109v1.Google Scholar
Serre, J. P., Trees (Springer, Berlin, 2003).Google Scholar