Subgroup Theorems for Groups Presented by Generators and Relations

Abstract

A problem that arises in every group is the determination of the structure of all subgroups contained in it. Early investigations of the special case of free groups culminated in the Reidemeister-Schreier theorem [1]2 which gives generators and relations for any subgroup of a group defined by generators and relations in terms of certain knowledge of its left cosets. In ?1, which collects together elementary definitions and results for use in the remainder of the paper, we show that the problem of Reidemeister and Schreier is exactly equivalent to that of giving free generators for a free group. We give a new solution to this problem (cf. Theorems 2.11, 2.25, and 2.27 of ?2) which permits application to questions of the structure of subgroups of groups which are free products or free products with one identified subgroup (?3, 4, 6). In particular, it provides an immediate proof of the Kurosh subgroup theorem [2] which states that every subgroup of a free product G is itself a free product, whose are a free group and subgroups of conjugates of the of G. The Kurosh proof, which is combinatorial in nature and utilizes a complicated double transfinite induction, was replaced by Baer and F. Levi with a topological proof [3] which represents a generalization of the second Johannson proof [4] of the ReidemeisterSchreier theorem. Their methods enable them to sharpen the theorem in an essential manner; namely, they provide us with a factorization with the largest possible factors which are subgroups of conjugates of of G. A further improvement of Kurosh's theorem was made by Takahasi [5] who established a relation between the number of of G and the number of of the subgroup. As proved in ?3, our statement of the Kurosh subgroup theorem includes all of these results, admitting in addition an explicit construction of generators for the subgroup. Contrary to all previous algebraic proofs, which have utilized the comparison of the length of a product of group elements with those of the in an essential manner, no cancellation arguments are made. Furthermore, although the statements are close to those of Takahasi, the removal of minimal conditions allows considerable freedom in the construction of generators. Recently, H. Neumann [6] has investigated theorems analogous to the Kurosh subgroup theorem for generalized free products with identified subgroups. By a simple improvement in our choice of generators, we obtain Theorem 4.08