ON PROFINITE GROUPS WHOSE POWER SUBGROUPS ARE CLOSED

We say that a profinite group G has Property S if for each integer n, there is a bound k such that every element of the nth power subgroup is a product of k nth powers of elements of G. Finitely generated profinite groups with Property S are completely determined by their group structure in a way conjectured by Hartley to be true of all finitely generated profinite groups; namely, every subgroup of finite index is open. We show that the class of finitely generated profinite groups with Property S is closed under forming extensions of its members and under taking subgroups of finite index. As a consequence, we note that all profinite groups of finite rank have Property S.