\documentclass[12pt]{article}

\usepackage{times}
\usepackage{graphics}
\usepackage{amssymb}
\usepackage{amsmath}

\def\Real{{\mathbb R}}
\def\be{\begin{equation*}}
\def\ee{\end{equation*}}
\def\tr{{\rm tr}}


\begin{document}

\begin{center}
{\bf \Large 250A Homework 3}\\[7mm]
{\small Due Monday October 25}\\[4mm]
\end{center}

\noindent
{\bf Question 1} 
Definition: ${\rm tor}(G)=\{g\in G|g^n=1 \mbox{for some } n\in {\mathbb N}\}$.
In addition a group $G$ is called torsion-free if ${\rm tor}(G)=\{1\}$.

Let $\Gamma$ be a group which contains a
torsion-free subgroup $\Gamma_0$ of index $n<\infty$. Show that
$\Gamma$ does not contain any element whose order is finite and
strictly larger than $n$.

\vspace{.4cm}\noindent
{\bf Question 2} 
Show that ${\rm tor}(G)\leq G$ if $G$ is abelian.

\vspace{.4cm}\noindent
{\bf Question 3} 
Show that the order of a permutation is the least common
multiple of the lengths of its disjoint cycles.

\vspace{.4cm}\noindent
{\bf Question 4} 
Find all normal subgroups of $S_4$.


\vspace{.4cm}\noindent
{\bf Question 5} Show that $K\leq H\leq G$ 
and $K\unlhd G$ $\Rightarrow$ $K\unlhd H$. 
Now, a permutation is called even if it can be written as
an even number of transpositions (e.g. (132)=(12)(13)
is even). The alternating group $A_n$ is the subgroup of
the symmetric group $S_n$ consisting of only even permutations.
Study the group $A_4$ to decide whether $K\unlhd H\unlhd G$ 
$\Rightarrow$ $K\unlhd G$.

\vspace{.4cm}\noindent
{\bf Question 6} Show that $Sl(n,{\mathbb R})\unlhd Gl(n,{\mathbb R})$.


 
\end{document}
