Groups whose prime graph on conjugacy class sizes has few complete vertices.
Let \(G\) be a finite group and let \(\mathrm{cs}(G)\) be the set whose elements are the sizes of the conjugacy classes of \(G\). In the analysis of the interplay between the structure of \(G\) and \(\mathrm{cs}(G)\) a key tool is played by the \textit{prime graph} of \(\mathrm{cs}(G)\), that is the graph \(\Gamma(G)\) whose vertices are the prime numbers dividing some element of \(\mathrm{cs}(G)\), and two (distinct) vertices \(p\) and \(q\) are adjacent in \(\Gamma(G)\) if and only if there exists an element in \(\mathrm{cs}(G)\) that is divisible by \(pq\). The graph \(\Gamma(G)\) tends to have many edges, in the sense that non-adjacency of two vertices \(p\) and \(q\) implies restrictions on the structure of \(G\). Define a vertex of a graph as \textit{complete} if it is adjacent to all the other vertices of the graph. One of the main results of this interesting paper is (Theorem A and Theorem 2.4): Let \(G\) be a finite group and assume that at most one vertex of \(\Gamma(G)\) is complete. Then \(G\) is solvable and \(G/F_2(G)\) is Abelian; in particular the Fitting length of \(G\) is at most \(3\). -- In addition, under the assumptions of Theorem A, if \(2\) is not a complete vertex, \(G'F(G)/F(G)\) is Abelian. Another result is Theorem C: Let \(G\) be a finite group and assume that no vertex of \(\Gamma(G)\) is complete. Then, up to an Abelian direct factor, \(G=G'H\), \(G'\) and \(H\) are Abelian groups of coprime order. Moreover \(G'\cap Z(G)=1\) and the prime divisors of \(|G'|\) (also of \(|H|\)) are pairwise adjacent vertices in \(\Gamma(G)\). The authors obtain also a full characterization of finite groups for which \(\Gamma(G)\) is a non-complete regular graph (Theorem D) and a characterization of the non-complete regular graphs which occur as \(\Gamma(G)\) for some finite group \(G\) (Corollary E). This nicely written paper concludes with a series of examples illustrating the aforementioned results.
scientific article; zbMATH DE number 6138033