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% 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}
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\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}. 
Actually we prove Theorem \ref{thm:3} for specified
tori  when we verify Theorem \ref{thm:1} for the simple groups of Lie type.
%\bigskip
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Prime graphs and Brauer characters,
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\end{document}
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