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. Ž Ž . . 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 Ž . category P P M M E E G, V V and, using the above generalization, show that if G is V V-perfect, then there exists a universal object in this category and its structure will be determined. Finally it is shown that any two V V-covering groups of a V V-perfect group are isomorphic and the structure of the unique generalized covering group of an arbitrary V V-perfect group is introduced. ᮊ 2001 Academic Press
INTRODUCTIONThe study of varietal perfect groups is very useful in some group theoretical constructions. For instance, investigations of varietal perfect groups may be applied in the existence of V V-stem covers of a group. More w x precisely, Leedham-Green and Mckay 7 showed that if a group G and a variety V V satisfy this condition in that the image of the identity map under the homomorphismG is zero, then at least one V V-stem cover of G exists and vice versa. In other words the existence of V V-covering groups for G is equivalent to the above
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