%% Math 351 Final, Fall 2018
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\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2018} \\
\textbf{MATH 351, Final Examination} \\
\textbf{Friday, December 14}
\end{center}

\vfil

\noindent\textbf{Your Name}: \Blank
\vskip 0pt plus .3fil

\paragraph{Instructions:} Please 
show all work necessary to justify your answers.
Use the back of the preceding page if you need more space 
for scratch work.  There are 200 possible points distributed as below.
The parts marked with (*) will be used in computing the Midterm 1
subscore.  

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
I  & \hfil/\  30\hfil \Strut \\
\hline
II  & \hfil/\  30\hfil \Strut \\
\hline
III  & \hfil/\  30\hfil \Strut \\
\hline
IV  & \hfil/\  40\hfil \Strut \\
\hline
V  & \hfil/\  30\hfil \Strut \\
\hline
VI  & \hfil/\  40\hfil \Strut \\
\hline\hline
Midterm I Sub & \hfil/80\hfil \Strut \\
\hline\hline
Total   & \hfil/200\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\centerline{\it Have a peaceful and joyous holiday season!}

\vfil\eject

\begin{enumerate}

\item[I.]  Both parts of this
question deal with  $SL(2,\mathbb{Z})$, the set of $2\times 2$ integer matrices
of determinant 1, a group under matrix multiplication.  Let
$$H = \left\{A = \begin{pmatrix} a & b\\ c & d\end{pmatrix} \in SL(2,\mathbb{Z}) : 
c \equiv 0 \bmod 5\right\}.$$
\begin{enumerate}
\item[(A)] (*) (15)  Is $H$ a subgroup of $SL(2,\mathbb{Z})$?  Why or why not?

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\item[(B)] (*) (15)  Is the cyclic subgroup generated by 
$$A = \begin{pmatrix} 1 & 0 \\ 5 & 1\end{pmatrix}$$
in $SL(2,\mathbb{Z})$ finite or infinite?  Explain.
\end{enumerate}

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\item[II.] Let $G = \langle a\rangle$ be a cyclic group of order $100$.
\begin{enumerate}
\item[(A)]  (*) (10)  How many different generators does $G$ have?

\vskip 2.0truein

\item[(B)]  (*) (10)  What is the order of the element $a^{30}$ in $G$?

\vskip 2.0truein

\item[(C)]  (*)  (10)  Suppose you know that a subgroup $H$ of $G$ contains both 
$a^{30}$ and $a^{56}$. What can you say about the order of $H$?  

\end{enumerate}
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\item[III.]
\begin{enumerate}
\item[(A)] (10)  Let $\alpha : G \to H$ be a group homomorphism.  Show that $\alpha(G)$ is a 
subgroup of $H$.

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\item[(B)] (20)  {\it State and prove} the First Isomorphism Theorem for groups.

\end{enumerate}

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\item[IV.]  All parts of this question refer to the group $G$ of
order $8$ whose operation table is given below:
\[
\begin{array}{cccccccccc}
\cdot & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 \\ \hline
g_1 & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 \\
g_2 & \vline & g_2 & g_5 & g_4 & g_7 & g_6 & g_1 & g_8 & g_3 \\
g_3 & \vline & g_3 & g_8 & g_5 & g_2 & g_7 & g_4 & g_1 & g_6 \\
g_4 & \vline & g_4 & g_3 & g_6 & g_5 & g_8 & g_7 & g_1 & g_2 \\
g_5 & \vline & g_5 & g_6 & g_7 & g_8 & g_1 & g_2 & g_3 & g_4 \\
g_6 & \vline & g_6 & g_1 & g_8 & g_3 & g_2 & g_5 & g_4 & g_7 \\
g_7 & \vline & g_7 & g_4 & g_1 & g_6 & g_3 & g_8 & g_5 & g_2 \\
g_8 & \vline & g_8 & g_7 & g_2 & g_1 & g_4 & g_3 & g_6 & g_5
\end{array}
\]

\begin{enumerate}
\item[(A)] (*) (5) What is the inverse of the element $g_2$?

\vskip 1.0truein

\item[(B)] (*) (5) What elements are in the subgroup $\langle g_3\rangle$?

\vskip 1.0truein

\item[(C)] (*) (5) Is the subgroup $\langle g_3\rangle$ normal in $G$?  Why or why not?

\end{enumerate}

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\noindent
Here is the group table of $G$ again so you don't need to flip back to the last page:
\[
\begin{array}{cccccccccc}
\cdot & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 \\ \hline
g_1 & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 \\
g_2 & \vline & g_2 & g_5 & g_4 & g_7 & g_6 & g_1 & g_8 & g_3 \\
g_3 & \vline & g_3 & g_8 & g_5 & g_2 & g_7 & g_4 & g_1 & g_6 \\
g_4 & \vline & g_4 & g_3 & g_6 & g_5 & g_8 & g_7 & g_1 & g_2 \\
g_5 & \vline & g_5 & g_6 & g_7 & g_8 & g_1 & g_2 & g_3 & g_4 \\
g_6 & \vline & g_6 & g_1 & g_8 & g_3 & g_2 & g_5 & g_4 & g_7 \\
g_7 & \vline & g_7 & g_4 & g_1 & g_6 & g_3 & g_8 & g_5 & g_2 \\
g_8 & \vline & g_8 & g_7 & g_2 & g_1 & g_4 & g_3 & g_6 & g_5
\end{array}
\]
\begin{enumerate}
\item[(D)] (*) (5) What is the center of $G$, that is, the subgroup $Z(G)$?

\vskip 1.0truein

\item[(E)] (20) Construct the group table for the factor group $G/Z(G)$.
To which ``standard'' group is this isomorphic?

\end{enumerate}

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\item[V.]   
\begin{enumerate}
\item[(A)] (15)  Up to isomorphism, how many different \emph{abelian} groups 
of order $600$ are there?  List one group from each isomorphism class.
\vskip 3.5truein
\item[(B)] (15)  Up to isomorphism, how many different groups of order $2018$ are
there?  List one group from each isomorphism class.  (Hint:  This is \emph{not} a long list!) 
\end{enumerate}

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\item[VI.] 
\begin{enumerate}
\item[(A)] (20) Use the Sylow Theorems to show that there are no simple groups
of order $100$.  
\vskip 3.0truein
\item[(B)] (20)  How many different Sylow $5$-subgroups does the alternating group 
$A_5$ have?

\end{enumerate}

\end{enumerate}

\end{document}


