On groups admitting a disconnected common divisor graph
Let \(G\) be an arbitrary group and \(A\) be a subgroup of \(G\). Accordingly with a paper by \textit{I. M. Isaacs} and \textit{C. E. Praeger} [ibid. 159, No. 1, 158-175 (1993; Zbl 0815.20002)], the common divisor graph \(\Gamma\) associated to the pair \((G,A)\) has as vertices the subdegrees of \((G,A)\), with an edge joining two vertices if and only if they are not coprime. There it was shown that \(\Gamma\) has at most two connected components. The paper under review gives structure theorems on the behaviour of \(G\) and \(A\) in both the connected and the disconnected case. Some examples are also provided.
scientific article; zbMATH DE number 1051578