On power subgroups of profinite groups

Abstract

In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup Gn , generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements xl, ... , xm, then there is a function f(m, n) such that if every word in x?l of length 1, we denote the subgroups of G generated by all nth powers an, a E G. A. Shalev conjectured that for any n the subgroup Gn is closed in G. This is the same as saying that for arbitrary integers m > 1, n > 1 there exists an integer N = N(m, n) such that in an arbitrary m-generated finite group G every product of nth powers of elements of G can be represented in the form a, **N where ai E G, 1 < i < N. Let us show, for example, that the existence of a function N(m, n) implies that Gn is closed. The subset M = {an . . *aN: al, ... , aN E G} of G is closed as the image of the compact G x ... x G under the continuous map (al, ., aN) -a an ... an Now we can consider the finite nilpotent group G/H, where H is an arbitrary open subgroup of G and so we have GnH = MH, which implies that Gn lies in the closure of M. Hence Gn = M. In this paper we prove Shalev's Conjecture when G is a pro-(finite nilpotent) Received by the editors February 8, 1994. 1991 Mathematics Subject Classification. Primary 20E18; Secondary 20F05. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page