On the connected components of the prime and Sylow graphs of a finite group
Let \(G\) be a finite group and let \(\pi(G)\) be the set of prime divisors of \(|G|\). The prime graph (or Gruenberg-Kegel graph) \(\Gamma_p(G)\) is constructed as follows: the vertices of \(\Gamma_p(G)\) are elements of the set \(\pi(G)\) and two vertices \(p,q\in\pi(G)\) are joined by an edge if and only if \(G\) contains an element of order \(pq\). This notion was introduced by Gruenberg and Kegel in the 1970s. One of the first results concerning prime graphs of groups was the unpublished Gruenberg-Kegel theorem, which described the structure of a finite group whose prime graph is disconnected. This theorem was published later in a paper by \textit{J. S. Williams} [J. Algebra 69, 487--513 (1981; Zbl 0471.20013)] In the following years, many different results about the structure of a prime graph of a finite group were obtained. In particular, the diameter of this graph is at most \(5\) [\textit{M. S. Lucido}, J. Group Theory 2, No. 2, 157--172 (1999; Zbl 0921.20020)]. The Sylow graph \(\Gamma_s(G)\) of a group \(G\) is a graph with \(V(\Gamma_s(G)) = \pi(G)\) and two vertices \(p\), \(q\) in form an edge of if either \(q\in\pi(N_G(G_p)/G_pC_G(G_p))\) or \(p\in\pi(N_G(G_q)/G_qC_G(G_q))\), where \(G_p\) is a Sylow \(p\)-subgroup of \(G\). The Sylow graph has been studied by \textit{L. S. Kazarin} et al. in [Isr. J. Math. 186, 251--271 (2011; Zbl 1280.20027)]. They proved that if \(G\) is an almost simple group, then the graph \(\Gamma_s(G)\) is connected and has diameter at most \(5\). The connectivity of \(\Gamma_s(G)\) was proved involving the classification of finite simple groups. \textit{F. G. Russo} in [``Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group'', Adv. Pure Math. 2, No. 6, 391--396 (2012, \url{doi:10.4236/apm.2012.26058 })] pointed out that if \(G\) is an almost simple finite group, then diameters of graphs \(\Gamma_p(G)\) and \(\Gamma_s(G)\) are at most \(5\). In this connection, he formulated the question: Is it possible to find relations between the connected components of \(\Gamma_p(G)\) and those of \(\Gamma_s(G)\), when \(G\) is a finite group? In the paper under review, the author proves that if \(G\) is a finite group, then either \(\Gamma_s(G)\) or \(\Gamma_p(G)\) is connected. In particular, the join of vertices of connected components of one of these graphs is the set of vertices of the unique connected component of the other graph. Thus, in the author's opinion, it gives an answer to Russo's question.
scientific article; zbMATH DE number 7489065
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