A Subgroup Theorem for Free Products of Pro-Finite Groups

The notion of a free product of pro-finite groups has some important applications in the theory of algebraic number fields (see [3]). In this connection, it is interesting to get some knowledge about the subgroups of such a free product. The aim of this paper is to show a theorem for the open subgroups of a free pro-finite product, which is an analog of Kurosh’s wellknown subgroup theorem for the free products of discrete groups in the usual sense. By K we denote a class of finite groups, which is closed under formation of subgroups, factorgroups and group extensions (c.g., all finite groups, the solvable groups, the p-groups, etc.) By a puo-@roup 6 we understand a projective limit 6 7 I;m Gi of groups Gi E (5. Let 6, , E E 81, be a family of pro-K-groups. A family of homomorphisms 7% : Cc,, + -5 into a pro-K-group $j is called convergent, if every open subgroup of $ contains almost every (i.e., up to a finite number) of the images ~~(05~). This condition is clearly empty, if VI is a finite index set. The free pro-Kpvoducc of the pro-K-groups 6, is now defined as to be a pro-K-group (5 =: JJ&,[ . 6, , together with a convergent family of homomorphisms