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.

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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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