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\begin{document}

\begin{center}
{\bf\large  ALGEBRA TEST \#2}
\end{center}

\vspace{1in}

This exam is due Wednesday, November 14.
You may use your book, but you may not
communicate with others concerning the exam.
In order to receive full credit your answer must be 
{\bf complete}, {\bf legible} and {\bf correct}. 

\vfill

\begin{center}
I have neither given nor received aid on this exam.
\end{center}

\vspace{.5in}

\begin{flushright}
{\bf Name:\underline{\hspace{2.5in}}}
\end{flushright}

\break

1. Let $\m a = \lb G/H; G\rb$ be the $G$-set of left
cosets of $H$ under the action of left multiplication.
Show that $Aut(\m a)\cong N_G(H)/H$.

\vfill

2. Find all (isomorphism types of) 
groups of order $18$.

\vfill

3. Show that if $|G|=24$ and $G$ has no normal Sylow
subgroup, then $G\cong S_4$.

\vfill

4. Do Exercise 4.5.27 from the book.
%Identify $S_{p^2}$ with the group of all permutations
%of the ring ${\BZ}_{p^2}$. Show that $S_{p^2}$
%has a Sylow $p$-subgroup consisting of polynomial functions
%over this ring. (Hint: Show that the collection of all 
%$a_0 + a_1x + \cdots + a_nx^n\in {\BZ}_{p^2}[x]$
%for which $a_1\equiv 1\pmod{p}$ and $a_i\equiv 0\pmod{p}$
%for $i\geq 2$ is a Sylow $p$-subgroup.)

\vfill

\end{document}
