Products of powers in finite simple groups

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.