$p$-subgroup PGroup 2003-10-15 01:29:47 2004-12-12 14:57:43 Definition Sylow $p$-subgroup Sylow p-subgroup $p$-group p-group \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} Let $G$ be a finite group with order $n$, and let $p$ be a prime integer. We can write $n=p^k m$ for some $k,m$ integers, such that $k$ and $m$ are coprimes (that is, $p^k$ is the highest power of $p$ that divides $n$). Any subgroup of $G$ whose order is $p^k$ is called a Sylow $p$-subgroup. While there is no reason for Sylow $p$-subgroups to exist for any finite group, the fact is that all groups have Sylow $p$-subgroups for every prime $p$ that divides $|G|$. This statement is the First Sylow theorem When $|G|=p^k$ we simply say that $G$ is a $p$-group.