Note on a theorem of Magnus

Source 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 t x 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.