Powers in finite groups. (Q664242)

If \(G\) is a group and \(n\in\mathbb N\) define \(G^n=\langle g^n\mid g\in G\rangle\). In this paper (which should be seen as a sequel to the seminal work of the authors [Ann. Math. (2) 165, No. 1, 171-238, 239-273 (2007; Zbl 1126.20018)]) it is proved that, if \(n,d\in\mathbb N\) and \(G\) is a \(d\)-generator finite group, then there exists \(f=f(n,d)\) such that every element of \(G^n\) can be written as \(g_1^n\cdot g_2^n\cdots g_f^n\) (\(g_i\in G\)). Together with the positive solution of the restricted Burnside problem this implies that, if \(H\) is a finitely generated profinite group, then \(H^n\) is open in \(H\) for every \(n\in\mathbb N\) and that (by Schreier's formula), if \(G\) is a \(d\)-generated finite group, then there exists \(\delta=\delta(n,d)\) such that \(G^n\) is generated by at most \(\delta\) \(n\)-th powers of elements of \(G\).