ON PROFINITE GROUPS WHOSE POWER SUBGROUPS ARE CLOSED

ABSTRACT

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.

5. ACKNOWLEDGMENTS

The second author wishes to thank the University of Alabama for their financial support while the work for this paper was being done.