A Remark on Generalized Covering Groups

TL;DR

This paper investigates the existence of covering groups within the variety of nilpotent groups of class at most c, demonstrating that certain finite cyclic groups lack such coverings when their orders are not coprime.

Contribution

It provides a specific non-existence result for ${\

Findings

Finite cyclic groups with non-coprime orders have no ${\cal N}_c$-covering groups for c ≥ 2.

The result indicates limitations in generalizing Wiegold's and Haebich's theorems to nilpotent groups of higher class.

The paper offers insights into the structure of covering groups within a particular variety of nilpotent groups.

Abstract

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 $c\geq 2$.