Groups with exponents I. Fundamentals of the theory and tensor completions

We revise R. Lyndon's notion of group with exponents [1]. The advantage of the revised notion is that, in the case of abelian groups, it coincides with the notion of a module over a ring. Meanwhile, the abelian groups with exponents in the sense of Lyndon form a substantially wider class. In what follows we introduce basic notions of the theory of groups with exponents; in particular, we present the key construction in the category of groups with exponents, that of tensor completion. The main results of the article are exposed in [2]; the notions of free A-group and free product of A-groups can be found in [3].