The Category (G, ) and Generalized Covering Groups,

The properties of covering and universality between the central extensions and the structure of a covering group of perfect groups have been generalized by S. Kayvanfar and M. R. R. Moghaddam (1997, Indag. Math. N.S.8(4), 537–542) to the variety of groups defined by a set of outer commutator words. In this paper we generalize the above results to any variety of groups. Then we introduce the category (G, ) and, using the above generalization, show that if G is -perfect, then there exists a universal object in this category and its structure will be determined. Finally it is shown that any two -covering groups of a -perfect group are isomorphic and the structure of the unique generalized covering group of an arbitrary -perfect group is introduced.