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Israel Journal of Mathematics

, Volume 96, Issue 2, pp 469–479 | Cite as

Products of powers in finite simple groups

Article

Abstract

LetG be a group. For a natural numberd≥1 letG d denote the subgroup ofG generated by all powersa d ,aG.

A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromG d can be represented asa 1 d ...a N d ,a i G. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsG d are closed and that an arbitrary subgroup of finite index inG is closed. In [5,6] the first author proved the existence of such a function for nilpotent groups and for finite solvable groups of bounded Fitting height.

Another interpretation of the existence ofN(m, d) is definability of power subgroupsG d (see [10]).

In this paper we address the question for finite simple groups. All finite simple groups are known to be 2-generated. Thus, we prove the following: THEOREM:There exists a function N=N(d) such that for an arbitrary finite simple group G either G d =1 orG={a 1 d ...a N d |a i G}.

The proof is based on the Classification of finite simple groups and sometimes resorts to a case-by-case analysis.

Keywords

Weyl Group Simple Root Arbitrary Element Dynkin Diagram Chevalley Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 1996

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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