Israel Journal of Mathematics

, Volume 96, Issue 2, pp 469–479

Products of powers in finite simple groups

Article

Abstract

LetG be a group. For a natural numberd≥1 letGd denote the subgroup ofG generated by all powersad,aG.

A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromGd can be represented asa1d...aNd,aiG. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsGd 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 subgroupsGd (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 Gd=1 orG={a1d...aNd|aiG}.

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

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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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