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\begin{document}
{\bf Math 177 HW 4 Due Friday, Feb 16 at noon}
\begin{enumerate}
\setcounter{enumi}{25}
\item Suppose $G$ is a group where every element has order 2. Show that $G$ is Abelian.

\item\label{all groups of order 6} Find all groups of order 6.
\item \label{GL(2, Z2)} Suppose $p$ is a prime. Let $G$ be the set of all 2 by 2 matrices with entries which are integers mod $p$ that have determinant not equal to zero mod $p$.  
\begin{enumerate}
\item Using matrix multiplication as the operation (and, again, doing all integer arithmetic mod $p$), show that $G$ is a group. This group is called $GL(2, 
\mathbb Z/p \mathbb Z)$. The ``GL'' stands for ``general linear'' group, the ``2'' for the size of the matrices, and finally $\mathbb Z/p\mathbb Z$ are the integers mod $p$.  
\item Let $SL(2, \mathbb Z/p \mathbb Z)$ be the subset of $GL(2, \mathbb Z/p \mathbb Z)$ of matrices with determinant equal to 1 mod $p$. Show that $SL(2, \mathbb Z/p \mathbb Z)$ is a subgroup of $GL(2, 
\mathbb Z/p \mathbb Z)$. The group $SL(2, \mathbb Z/p \mathbb Z)$ is called the ``special linear'' group.
\item Show that $SL(2, \mathbb Z/2 \mathbb Z)=GL(2, \mathbb Z/2 \mathbb Z)$ and has order 6.
Where does this group appear in your answer to Exercise \ref{all groups of order 6}?
\item What are the orders of $SL(2, \mathbb Z/p \mathbb Z)$ and $GL(2, \mathbb Z/p \mathbb Z)$ for any prime $p$? This might be hard. Here are the orders for small $p$, from which you might be able to guess at the answer and then use that to help prove the general result. These results were obtained by writing a computer program to generate all possible matrices for each $p$ and then count the ones that had non-zero determinant  or determinant 1.
$$\begin{array}{rrr}
p&|SL(2, \mathbb Z/p \mathbb Z)|&|GL(2, \mathbb Z/p \mathbb Z)|\\
\hline
2&6&6\\
3&24&48\\
5&120&480\\
7&336&2016\\
11& 1320& 13200\\
13&2184& 26208\\
17&4896& 78336\\
19&6840& 123120
\end{array}
$$
\end{enumerate}

\item Suppose that $H$ is a subgroup of the group $G$.
\begin{enumerate}
\item Declare that $x, y\in G$ are {\it equivalent}, written $x\sim y$, if $x^{-1}y\in H$. Show that this is an equivalence relation on $G$.
\item Show that the equivalence classes are exactly the right cosets of $H$
\item What equivalence relation will have as its equivalence classes the left cosets of $H$?
\end{enumerate}

\item Show that half the permutations in $S_n$ are even and half are odd. Thus  $|A_n|=n!/2$.

\item Show that $S_n$ can be generated by $(1\,2)$ and the $n$-cycle $(1\,2\,3\dots \,n)$.

\item Show that every subgroup of an Abelian group  is normal.

\item If $n$ is a natural number, let $n\mathbb Z$ denote all integer multiples of $n$.
\begin{enumerate}
\item Show that $n\mathbb Z$ is a normal subgroup of $\mathbb Z$.
\item What are the right cosets of $n\mathbb Z$ and what is the index of $n\mathbb Z$ in $\mathbb Z$?
\item What is $n\mathbb Z\cap m\mathbb Z$?
\item Is every subgroup of $\mathbb Z$ of the form $n\mathbb Z$ for some $n$?
\end{enumerate}
\end{enumerate}
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