A Remark on Generalized Covering Groups

Details A Remark on Generalized Covering Groups A Remark on Generalized Covering Groups

2011-04-03

arxiv.org/abs/1104.0397v1

Indian Journal of Pure and Applied Mathematics, 29:7 (1998) 317-322

Mathematics

Group Theory

6 pages

Scientific paper

Let ${\cal N}_c$ be the variety of nilpotent groups of class at most $c\ \ (c\geq 2)$ and $G=Z_r\oplus Z_s $ 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 ${\cal N}_c$-covering group for every $c\geq 2$. 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 {\it not} be generalized to the variety of nilpotent groups of class at most $c\geq 2$.

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