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\noindent
\parbox{2in}{\bf Math 3110} 
\hfill {\Large \bf Test \#3} \hfill
\parbox{2in}{\bf \hfill April $7^{\mathrm{th}}$, 2021}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!}

\vspace{0.2in}

\noindent {\large 1. (15 points)} Getting things in order\dots
\begin{enumerate}[(a)]
   \item Let $G = Q \times A_4$ where $Q = \{ \pm 1, \pm i, \pm j, \pm k\}$ is the quaternion group and $A_4$ is the group of even permutations in $S_4$ (i.e., $A_4 = \{ (1), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23) \}$).

\vspace{0.1in}

The order of $G$ is $|G|=$ \underline{\hspace*{1in}}

\vspace{0.1in}

What is the largest element order in $Q \times A_4$? {\bf Give an example} of such an element.

\vspace{1in}

\item Let $G$ be a group of order 50 with subgroups $H$, $K$, and $L$. In addition, suppose the order of $H$ is 5, the order of $L$ is 2, and that $H \subseteq K \subseteq G$. 

\vspace{0.15in}

\noindent
What is/are the possible order(s) of $K$?

\vspace{0.5in}

\noindent
What can we conclude about $H \cap L$?

\vspace{0.5in}

\end{enumerate}


\noindent {\large 2. (15 points)} Let $\varphi:G \to H$ be a homomorphism.\\ \force \qquad \qquad $\longrightarrow$ State the definition a homomorphism.\\ \force \qquad \qquad $\longrightarrow$ State the definition of the kernel: $\mathrm{ker}(\varphi)$.\\ \force \qquad \qquad $\longrightarrow$ Then prove that $\mathrm{ker}(\varphi)$ is a normal subgroup of $G$. \hfill
{\it Note:} Prove it is a subgroup \underline{and} that it is normal in $G$.


\vfill

\newpage

\newpage
\noindent {\large 3. (15 points)} Consider $K = \{ 1,x^2,x^4 \}$ in $D_6 = \langle x,y \;|\; x^6=1,y^2=1,xyxy=1 \rangle = \{ 1,x,\dots,x^5, y,xy,\dots, x^5y\}$. 

{\it Note:} It can be shown that $K$ is a normal subgroup of $D_6$ (just accept this for now).

\begin{enumerate}[(a)]
   \item  Quick questions about $\diagquotient{D_6}{K}$.
   
\vspace{0.15in}   

   The order of $\diagquotient{D_6}{K}$ is \underline{\hspace*{0.5in}}.
   \hfill
   List the distinct elements of $\diagquotient{D_6}{K} = \Big\{$ \underline{\hspace*{2.25in}} $\Big\}$.\\
   \force \hfill {}[In terms of $K$, for example: ``$xK$''.] \  \  \force

\vspace{0.15in}   

   The identity of $\diagquotient{D_6}{K}$ is \underline{\hspace*{1in}}.
   \hfill
   $(xK)^{-1}=$ \underline{\hspace*{2in}}.\\
   \force \hfill {}[Simplify please.] \force

\vspace{0.15in}

  The order of $xK$ in $\diagquotient{D_6}{K}$ is \underline{\hspace*{1in}}. \hfill List the contents of $xK = \Big\{$ \underline{\hspace*{1.75in}} $\Big\}$.

\vspace{0.05in}

Scratch work:

\vspace{1.75in}   

\item Let $H = \{ 1, x^3, y, x^3y\}$. While $H$ is a subgroup of $D_6$, it is {\bf not} a normal subgroup of $D_6$. Prove it isn't normal.

\vspace{1.5in}

\end{enumerate}

 \noindent 
 {\large 4. (10 points)} Consider $\diagquotient{\mathbb{Z}_{12}}{H}$ where $H=\langle 3 \rangle = \{0,3,6,9\}$.\\
 \force \qquad \qquad $\longrightarrow$ List all of the cosets of $H$ (and their contents) in $\mathbb{Z}_{12}$.\\ 
 \force \qquad \qquad $\longrightarrow$ Then make a Cayley table for this quotient group. 
 
 
 \vfill
 
 What is the order of $8+H$ in $\mathbb{Z}_{12}/H$?
 
 




\newpage
\noindent {\large 5. (15 points)} Something is terribly, horribly wrong!

\begin{enumerate}[(a)]

\item Let $H \triangleleft \mathbb{Z}_{100}$. Why is $\diagquotient{\mathbb{Z}_{100}}{H} \cong \mathbb{Z}_{5} \times \mathbb{Z}_5$ impossible?

\vfill

\item Let $\varphi:A_4 \to Q$ be a homomorphism (where $A_4$ and $Q$ are the same as in problem 1(a)). Why can't $\varphi$ be onto?  

\vfill

\item Recall $H = \{1,x^3,y,x^3y\}$ (a subgroup of $D_6$) from problem 3(b). Suppose $\varphi:D_6 \to \mathbb{Z}$ is a homomorphism. Why can't $\mathrm{ker}(\varphi)=H$?

\vfill

\end{enumerate}


\noindent {\large 6 (10 points)} Let $H$ be a subgroup of $G$ where $G$ is an Abelian group. First, prove that $H$ is in fact a normal subgroup. Then, prove that $\diagquotient{G}{H}$ is Abelian.

\vfill


\newpage
\noindent {\large 7. (20 points)} Finite Abelian Groups

\begin{enumerate}[(a)]
   \item List all of the non-isomorphic Abelian groups of order $100 =2^2 5^2$. Circle any that are cyclic.

\vfill

   \item How many non-isomorphic Abelian groups of order 449,878,000 are there?

   {\it Note:}  449,878,000 $=2^4  \cdot 5^3  \cdot 11^3 \cdot 13^2$ and there are 5 non-isomorphic Abelian groups of order 16 $=2^4$. \smiley

\vfill

   \item Are the groups $\mathbb{Z}_{6} \times \mathbb{Z}_{10} \times \mathbb{Z}_{10}$ and $\mathbb{Z}_{12} \times \mathbb{Z}_{50}$ isomorphic? Explain your answer.

\vfill

   \item What is the largest order among elements of $\mathbb{Z}_{6} \times \mathbb{Z}_{9} \times \mathbb{Z}_{15}$? Explain you answer.

\vspace{0.75in}
\end{enumerate}

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