Non-Abelian Sylow subgroups of finite groups of even order
The purpose of this paper is to prove the following main theorem: Every non-Abelian Sylow subgroup of a finite group of even order contains a non-trivial element which commutes with an involution. This theorem, announced by the authors [in Electron. Res. Announc. Am. Math. Soc. 4, No. 12, 88-90 (1998; Zbl 0923.20013)], is closely related to the prime graphs of finite groups. The prime graph \(\Gamma(G)\) 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 \(r\) are joined by an edge if and only if there exists an element of order \(pr\) in \(G\). Let \(d_G(p,r)\) be the distance between two vertices \(p\) and \(r\) in \(\Gamma(G)\). The authors define \(d_G(p,r)=\infty\) if there is no path between \(p\) and \(r\). In terms of prime graphs, the main theorem is restated as Theorem 1: If \(G\) is a finite group of even order, \(p\) is a prime divisor of \(|G|\) and \(d_G(p,r)\geq 2\) then Sylow \(p\)-subgroups of \(G\) are Abelian. For the distance \(d_G(p,r)\), the authors prove Theorem 2: If G is a finite group of even order and \(p\) is a prime divisor of \(|G|\) then \(d_G(p,r)\in\{1,2,3,\infty\}\). If G is a finite non-Abelian group from Theorem 2 then \(d_G(p,r)\in\{1,2,\infty\}\) (Corollary 2). Note that Theorem 2 and Corollary 2 are also proved by \textit{M. S. Lucido} [see Lemma 5 and Proposition 7 in J. Group Theory 2, No. 2, 157-172 (1999; Zbl 0921.20020)]. The authors also prove the following Theorem 3: If \(G\) is a finite group of even order, \(p\) is a prime divisor of \(|G|\) and \(d_G(p,r)\geq 2\) then either \(G\) is solvable or \(G\) has a chain of normal subgroups \(K\subseteq H\subseteq G\) such that \(G/H\) and \(K\) are solvable and \(H/K\) is a non-Abelian simple group. Theorem 3 is an analog of the Gruenberg-Kegel theorem on the general structure of finite groups with disconnected prime graphs (see Theorem A in the paper by \textit{J. S. Williams} [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)]).
scientific article; zbMATH DE number 1443586