Note on a theorem of Magnus

Abstract

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.