On Some Lemmas in the Theory of Groups

Group-theoretic constructions like the one in Corollary 2 of the preceding paper have been used by several authors as a tool for the theory of abstract groups.* In this paper we shall show how the lemmas, proved by these authors on group constructions of that type, can be proved by means of simple topological reasonings on 2-dimensional complexes in the plane. The resulting proofs for those lemmas are shorter and of much clearer construction than the original proofs, but nevertheless liot essentially different. It is this 2-dimensional method of proof and not any original result which justifies the publication of this paper. The lemmas in section 1 of this paper give the connection between abstract groups and 2-cells, and are reminiscent of Dehn's theory of the Gruppenbild. Their proof is only sketched. In section 2 the lemmas proved by the authors cited are restated in a generalized form in 3 theorems with 2 corollaries. Nowhere in this paper is a restriction placed on the number (power of infinity) of any set of generators or of relations used.