A generalization of prime graphs of finite groups
The prime graph \(\Gamma(G)\) of a finite group \(G\) is the graph whose the vertex set is the set of prime divisors of \(|G|\), and two distinct vertices \(p\) and \(q\) are joined by an edge if and only if there exists a cyclic subgroup of order \(pq\) in \(G\). In this paper, the authors introduce some generalized prime graphs of finite groups, in which the condition ``being cyclic'' is replaced by other ones. They investigate some properties of such graphs, especially solvable graphs. The solvable graph \(\Gamma_{sol}(G)\) of a finite group \(G\) is the graph whose vertex set is the set of prime divisors of \(|G|\), and two distinct vertices \(p\) and \(q\) are joined by an edge if and only if there exists a solvable subgroup \(H\) in \(G\) such that \(pq\) divides \(|H|\). In the main theorems, it is proved that the solvable graph of a (known) finite non-Abelian simple group is connected and incomplete. The proofs use the description of the connected components of prime graphs for finite simple groups (see the papers of \textit{J. S. Williams} [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)], the reviewer [Mat. Sb. 180, No. 6, 787-797 (1989; Zbl 0691.20013)] and \textit{N. Iiyori} and \textit{H. Yamaki} [J. Algebra 155, No. 2, 335-343 (1993; Zbl 0799.20016)]). Some applications of solvable graphs are also given.
scientific article; zbMATH DE number 1532967