Products of powers in finite simple groups
For a natural number \(d\) and a group \(G\), define \(G^d\) as the subgroup generated by all \(d\)-th powers of elements in \(G\). If it is true that there exists a function \(N=N(m,d)\) such that, for every \(m\)-generated finite group \(G\), an arbitrary element from \(G^d\) can be represented as \(a_1^d\cdots a_N^d\), \(a_i\in G\) then, for every finitely generated profinite group \(P\), all power subgroups \(P^d\) and all subgroups of finite index in \(P\) are closed. Using the Classification of the Finite Simple Groups the authors prove the existence of such a function \(N\) in the case when \(G\) is simple. Earlier this was done for soluble groups of bounded Fitting height [\textit{C. Martinez}, Bull. Lond. Math. Soc. 28, No. 5, 481-487 (1996; Zbl 0876.20016)]. Note that another proof, also using the Classification of the Finite Simple Groups, was obtained by \textit{J. Saxl} and \textit{J. S. Wilson} [Math. Proc. Camb. Philos. Soc. 122, No. 1, 91-94 (1997; see the following review Zbl 0890.20013)].
scientific article; zbMATH DE number 997601
scientific article