Characterization of varietal covering and stem groups

Abstract

Let A _m be the variety of abelian groups of exponent m, where mis a square free positive integer. It is shown that every group G has an A _m -covering 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 groups, 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 this variety some results of Hall's type will be proved.

Keywords:

• A _m -covering Group
• A _m -isologism
• variety of groups

1991 Mathematics subject classification:

• (Amer. Math. Soc): 20E10
• (Amer. Math. Soc): 20F99