A Remark on Generalized Covering Groups

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Let Nc be the variety of nilpotent groups of class at most c(c≥ 2) and G=Zᵣ⊕ Zₛ be the direct sum of two finite cyclic groups. It is shown that if the greatest common divisor of r and s is not one, then G does not have any Nc-covering group for every c≥ 2. This result gives an idea that Lemma 2 of J.Wiegold [6] and Haebich's Theorem [1], a vast generalization of the Wiegold's Theorem, can not be generalized to the variety of nilpotent groups of class at most c≥ 2.