Products of powers in finite simple groups |
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Authors: | C. Martinez E. Zelmanov |
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Affiliation: | 1. Departamento de Matemáticas, Universidad de Oviedo, 33.007, Oviedo, Spain 2. Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA
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Abstract: | LetG be a group. For a natural numberd≥1 letG d denote the subgroup ofG generated by all powersa d ,a∈G. 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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