Inverse limits of solvable groups

Abstract

In this paper we generalize to groups of Galois type some results of P. Hall on finite solvable groups [l; 2; 3]. We need, in a modified form, some results of van Dantzig: the definition of supernatural numbers (which are related to van Dantzig's universal numbers) and Theorem 5, which he proved for ordinary ?>-Sylow subgroups [6]. Lemmas 1 and 4 and the method of proof in Theorem 5 are due to Tate [S]. A topological group G is of Galois type if it is compact and totally disconnected. In any Galois type group the open normal subgroups form a neighborhood base at the identity. Every closed subgroup is the intersection of the open subgroups containing it [4]. Whenever M and N are open normal subgroups of G and NZ)M we shall write 0jv for the natural homomorphism of G/M onto G/N (these quotient groups are finite) and <£# for the natural homomorphism of G onto G/N. G is the inverse limit of the groups {G/N}, N ranging over the open normal subgroups of G. Conversely, the inverse limit of finite groups is of Galois type.