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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 maurice.chiodo@unine.ch
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Abstract

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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. 

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