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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publisher's TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you either view the HTML version or * %_ * retrieve the article in DVI, PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{6-NOV-1998,6-NOV-1998,6-NOV-1998,6-NOV-1998} \documentclass{era-l} %\documentstyle[12pt,amssymb]{amsart} \newtheorem*{thm}{Main Theorem} % \renewcommand{\thethm}{} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \newtheorem*{problem}{Problem} % \renewcommand{\theproblem}{} \newtheorem*{lemma}{Lemma} %\renewcommand{\thelemma}{} \theoremstyle{definition} \newtheorem*{example}{Example} %\renewcommand{\theexample}{} %\renewcommand{\theremark}{} \begin{document} \title[ NONABELIAN SYLOW SUBGROUPS]{Nonabelian Sylow subgroups of finite groups of even order} \author[N.~Chigira]{Naoki Chigira} \address{Department of Mathematical Sciences, Muroran Institute of Technology, Hokkaido 050-8585, Japan} \email{chigira@muroran-it.ac.jp} \author[N.~Iiyori]{Nobuo Iiyori} \address{Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi 753-8512, Japan} \email{iiyori@po.yb.cc.yamaguchi-u.ac.jp} \author[H.~Yamaki]{Hiroyoshi Yamaki} \address{Department of Mathematics, Kumamoto University, Kumamoto 860-8555, Japan} \email{yamaki@gpo.kumamoto-u.ac.jp} % General info \subjclass{Primary 20D05, 20D06, 20D20} \issueinfo{4}{12}{}{1998} \dateposted{November 10, 1998} \pagespan{88}{90} \PII{S 1079-6762(98)00051-1} \def\copyrightyear{1998} \copyrightinfo{1998}{American Mathematical Society} \date{October 20, 1997} \commby{Efim Zelmanov} \keywords{Sylow subgroups, prime graphs, simple groups} %\noindent {\it AMS 1991 subject classifications}: \begin{abstract} We have been able to prove that every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. The proof depends upon the consequences of the classification of finite simple groups. \end{abstract} \thanks{The third author was supported in part by Grant-in-Aid for Scientific Research (No. 8304003, No. 08640051), Ministry of Education, Science, Sports and Culture, Japan.} \maketitle %\vspace{48pt} The purpose of this note is to announce \cite{ciy}: %\bigskip \begin{thm} Every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. \end{thm} %\bigskip Let $G$ be a finite group and $\Gamma (G)$ the prime graph of $G$. $\Gamma (G)$ is the graph such that the 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 $n( \Gamma (G))$ be the number of connected components of $\Gamma (G)$ and $d_G(p,r)$ the distance between two vertices $p$ and $r$ of $\Gamma (G)$. It has been proved that $n( \Gamma (G)) \leq 6$ in \cite{williams}, \cite{iy4}, \cite{kondrat'ev}, \cite{iy3}. %\bigskip \begin{theorem}\label{thm:1} Let $G$ be a finite group of even order and $p$ a prime divisor of $|G|$. If $d_G(2,p) \geq 2$, then a Sylow $p$-subgroup of $G$ is abelian. \end{theorem} %\bigskip Theorem \ref{thm:1} is a restatement of the Main Theorem in terms of the prime graph $\Gamma (G)$. %\bigskip \begin{corollary} Let $G$ be a finite group of even order and $p$ a prime divisor of $|G|$. If $\Delta $ is a connected component of $\Gamma (G) - \{p \}$ not containing $2$, then a Sylow $r$-subgroup of $G$ is abelian for any $r \in \Delta $. \end{corollary} There is a certain relation between a subgraph $\Gamma(G)-\{p\}$ of $\Gamma(G)$ and Brauer characters of $p$-modular representations of $G$ (see \cite{ci1}). %\bigskip \begin{theorem}\label{thm:2} Let $G$ be a finite nonabelian simple group and $p$ an odd prime divisor of $|G|$. Then $d_G(2,p)=1$ or $2$ provided $d_G(2,p)< \infty$. \end{theorem} %\bigskip %\nopagebreak The significance of the prime graphs of finite groups can be found in \cite{c1}, \cite{ci1}, \cite{iiyori1}, \cite{iiyori2}, \cite{iy1}, \cite{iy2}, \cite{y2}, \cite{y3}. We apply the classification of finite simple groups (see \cite{c1}, \cite{c2}, \cite{ciy}, \cite{iy1}, \cite{iy4}, \cite{y1}). It has been proved that a minimal counterexample to Theorem \ref{thm:1} is a nonabelian simple group. We will give some examples of case by case analysis for finite simple groups. Theorem \ref{thm:1} holds true for the sporadic simple groups by Atlas of Finite Groups although we can find several typos in it. For a positive integer $k$ let $\pi (k)$ be the set of all prime divisors of $k$. Let $\pi_0=\{p\in \pi(G) \mid d_G(2,p)=1\}$. Then we do not have to think about primes in $\pi_0$ in order to give the proof of Theorem \ref{thm:1}. \begin{example} Let $G$ be the alternating group on $n$-letters and $p\in \pi (G)$. It is trivial that Theorem \ref{thm:1} holds true for $A_5$ and $A_6$. Assume that $n\geq 7$. If $p\leq n-4$, then $d_G(2,p)=1$. If $p\geq n-3$, then Sylow $p$-subgroups of $G$ are cyclic of order $p$. Thus Theorem \ref{thm:1} holds true for the alternating groups. \end{example} \begin{example} Let $G=PSL(n,q)$, $q\equiv 0\pod{2}$. Then $|G|=q^{n(n-1)/2} \prod\limits_{i=1}^{n-1} (q^{i+1}-1)d^{-1}$, $d=(n,q-1)$. Let $I_j$ be the $j \times j$ identity matrix. Put %\begin{equation*} \[t_k^{\prime}= \begin{pmatrix} I_k & 0 & 0\\ 0 & I_{n-2k} & 0\\ I_k & 0 & I_k \\ \end{pmatrix}.\] %\end{equation*} Then $t_k^{\prime}\ (1\leq k \leq r)$, where $r=[n/2]$, are representatives of the conjugacy classes of involutions in $SL(n,q)$. The centralizer of $t_k^{\prime}$ in $SL(n,q)$ is the set of all matrices of the form \[\begin{pmatrix} A & 0 & 0\\ H & B & 0\\ K & L & A\\ \end{pmatrix},\] where $(\operatorname{det}A)^2\operatorname{det}B=1$ and $A$ is a $k\times k$ nonsingular matrix. Denote $t_k$ the homomorphic image of $t_k^{\prime}$ in $PSL(n,q)$. Then $t_k$ $(1\leq k \leq r)$ are representatives of the conjugacy classes of involutions in $PSL(n,q)$. Let $C_k=C_G(t_k)$. Then \[ \pi (C_k)=\pi (2 \prod\limits_{i=1}^{n-2k}(q^i-1)/(q-1)d)\] and \[ \pi_0 = \pi(\prod\limits_{k=1}^r|C_k|)=\pi (2 \prod\limits_{i=1}^{n-2}(q^i-1)).\] Suppose $n\geq 4$. Then the only factors of $|G|$ to be considered are $(q^{n-1}-1)(q^n-1)$. There are maximal tori $T(A_{n-2})$ of order $(q^{n-1}-1)d^{-1}$ and $T(A_{n-1})$ of order $(q^n-1)/(q-1)d$. Let $p\in \pi (T(X))-\pi_{0}$, where $X=A_{n-1}$ or $A_{n-2}$. Let $P$ be a Sylow $p$-subgroup of $T(X)$. Then $d_G(2,p)=1$ or $P$ is a Sylow $p$-subgroup of $G$. Since $P$ is abelian, Theorem \ref{thm:1} holds true for $G=PSL(n,q)$, $n\geq 4$. Suppose that $n=3$. Then $|G|=q^3(q^2-1)(q^3-1)d^{-1}$ and there are three classes of maximal tori of orders \[(q-1)^2d^{-1}, \quad (q^2-1)d^{-1}, \quad (q^2+q+1)d^{-1}.\] We note that a torus of order $(q^2+q+1)d^{-1}$ is an isolated subgroup. If $q > 4$, then $d_G(2,r)=2$ for $r\in \pi (q+1)$. Let $R$ be a Sylow $r$-subgroup of $G$. Then $R$ is contained in a maximal torus of order $(q^2-1)d^{-1}$. If $q=4$, then $G=PSL(3,4)$ and $|G|=2^6\cdot3^2\cdot5\cdot7$. If $q=2$, then $G=PSL(3,2)$ and $|G|=2^3\cdot3\cdot7$. We have verified Theorem \ref{thm:1} for $n=3$. It is trivial that Theorem \ref{thm:1} holds true for $PSL(2,q)$. \end{example} \begin{theorem}\label{thm:3} Let $G$ be a simple group of Lie type and $T$ a maximal torus. Let $p\in \pi(T)-\pi_0$. Then $T$ contains a Sylow $p$-subgroup of $G$. \end{theorem} Theorem \ref{thm:3} is a corollary of Theorem \ref{thm:1}. 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