%% Math 351 Midterm 1, Fall 2018
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\markright{MATH 351, Midterm 2}

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\textbf{College of the Holy Cross, Fall Semester, 2018} \\
\textbf{MATH 351, Midterm 2} \\
\textbf{Friday, November 16}
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\vfil

\noindent\textbf{Your Name}: \Blank
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\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 100 possible points distributed as below.

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

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\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
I  & \hfil/\  25\hfil \Strut \\
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II  & \hfil/\  30\hfil \Strut \\
\hline
III  & \hfil/\  25\hfil \Strut \\
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IV  & \hfil/\  20\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
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\begin{enumerate}

\item[I.] Let $G = U(20)$ (where the operation is multiplication mod $20$), and $N = \langle 9\rangle$ in $G$.
\begin{enumerate}
\item[(A)] (5) How do you know is $N$ a normal subgroup of $G$?  

\vskip 2.0truein

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

\end{enumerate}

\vfil\eject

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

\vskip 3.0truein

\item[(B)] (10)  State the First Isomorphism Theorem for groups.

\end{enumerate}
\vfil\eject


\begin{enumerate}
\item[(C)] (10) Let $G = \mathbb{Z} \times \mathbb{Z}$ and $N = \{(a,2a) \mid a \in \mathbb{Z}\}$.  
Using the First Isomorphism Theorem, determine a group isomorphic to $G/N$.
\end{enumerate}

\vfil\eject

\item[III.]  Let $G$ be a group of order $14$.   
\begin{enumerate}
\item[(A)] (15)  Show that $G$ contains elements of order $2$ and elements of order $7$.  
You may use without proof any general facts we know that apply here.  
\vfil\eject
\item[(B)] (10)  Still assuming $G$ has order 14, any element $a$ of order $7$ generates
a normal subgroup.  If $b$ has order $2$, determine all possibilities for $bab = bab^{-1}$. 
\end{enumerate}

\vfil\eject

\item[IV.]  
\begin{enumerate}
\item[(A)] (10) Using the Fundamental Theorem, give a complete list of abelian groups of 
order $72$ up to isomorphism.
  
\vfil\eject

\item[(B)] (5) Let $G = \mathbb{Z}_4 \times \mathbb{Z}_{18}$.  To which group in your list
from part (A) is $G$ isomorphic?

\vskip 2.0truein

\item[(C)] (5) Let $G = \langle a\rangle$ be a cyclic group of order $72$.  Write
$a = b\cdot c$, where $b$ is a $2$-element of $G$ and $c$ is a $3$-element of $G$.  
\end{enumerate}

\end{enumerate}

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