Characterization of varietal covering and stem groups

Let A(m) be the variety of abelian groups of exponent m, where m is a squar e free positive integer. It is shown that every group G has an A(m)-coverin g group, which generalizes the work of I. Schur (1904) and M.R. Jones (1973 ). Also, similar to the work of Yamazaki (1964) in the variety of abelian g roups, if 1 --> A --> H --> G --> 1 is an A(m)-stem extension, then we show that H is a homomorphic image of an A(m)-covering group of G. Finally,in t his variety some results of Hall's type will be proved.