The prime graphs of groups with arithmetically small composition factors (Q6196059)

Let \(G\) be a finite group. The prime graph (or Gruenberg-Kegel graph) \(\Gamma_{G}\) of \(G\) has the vertex set \(\pi(G)\) and \(p,q \in \pi(G)\), \(p \not =q\), are connected by an edge if and only if \(G\) has an element of order \(pq\). A fundamental tool in the study of the prime graph of a solvable group is the so-called Lucido's Three Primes Lemma [\textit{M. S. Lucido}, J. Group Theory 2, No. 2, 157--172 (1999; Zbl 0921.20020)], which asserts that if \(p\), \(q\), and \(r\) are distinct prime divisors of the order of a solvable group \(G\), then \(G\) contains an element of order the product of two of these three primes or, in other words, \(\overline{\Gamma}_{G}\) (the complement of \(\Gamma_{G}\)) must be triangle free. In [op. cit.], such result is used to prove that if \(G\) is solvable, then \(\mathrm{diam}(G) \leq 3\). The results of Lucido have been refined in a work by \textit{A. Gruber} et. al. [J. Algebra 442, 397--422 (2015; Zbl 1331.20029)], where it is proved that a graph \(\mathcal{G}\) is the prime graph of a solvable group if and only if \(\overline{\mathcal{G}}\) is triangle free and 3-colorable. In the paper under review, the authors classify the prime graph of groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. The simple (nonabelian) groups \(G\) such that \(|\pi(G)|=3\) are called \(K_{3}\)-groups and they constitute the finite set \[ \mathcal{K}_{3}=\big \{ A_{5},\; A_{6},\; L_{2}(7),\; L_{2}(8),\; L_{2}(17),\; L_{3}(3), \; U_{3}(3),\; U_{4}(2) \big \}. \] Let \(\mathcal{T}\) be a set of distinct (isomorphism types of) nonabelian simple groups. A group is called \(\mathcal{T}\)-solvable if each of its composition factors is either abelian or isomorphic to a group in \(\mathcal{T}\). A first result proved in this paper is (see Theorem 1.3): Let \(G\) be a \(\mathcal{K}_{3}\)-solvable group, and suppose one of the following holds: (1) \(G\) is solvable; (2) \(G\) has exactly one nonabelian composition factor, which is \(L_{3}(3)\), \(U_{3}(3)\), or \(U_{4}(2)\); (3) \(G\) has at least two (not necessarily distinct) nonabelian composition factors. Then \(\overline{\Gamma}_{G}\) is triangle-free and 3-colorable. The paper then presents some theorems that characterize the prime graph of \(\mathcal{T}\)-groups for several \(\mathcal{T} \subseteq \mathcal{K}_{3}\).