Subgroups of finite index in profinite groups

Abstract

Let ~ be a class of finite groups. By this we understand that ~ is a class in the usual sense, which contains all groups of order 1, and contains, with every group G ~ , all isomorphic copies of G. By a pro-~ group, we mean a topological group isomorphic to an inverse limit of groups in ~, viewed as a topological group in the usual way. If ~ is closed under taking homomorphic images, this is equivalent to saying that G is a compact totally disconnected Hausdorff topological group such that G/N~g for every open normal subgroup N of G. We write g* for the class of all pro-~ groups. It seems to be unknown whether every subgroup of finite index in a finitely generated profinite group is open. Here we say that a profinite group is finitely generated, if it has a dense subgroup which is finitely generated in the algebraic sense. The answer is known to be affirmative if ~ is the class 919l of finite abelian-by-nilpotent groups (Anderson [1]) or the class of finite supersoluble groups (Oltikar and Ribes [6]). I am indebted to L. Ribes for bringing these results to my attention, and for several stimulating discussions. We generalize these results as follows. For an integer l> 1, let 91l denote the class of all finite groups G which have a series