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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Sept. $15^\mathrm{th}$, 2023}

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

\vspace{0.05in}

\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

\vspace{0.2in}

\begin{description}
\item[\#1] A function problem
   \begin{enumerate}[(a)]
   \item Let $f : \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x)=|2x+1|+3$. 
      \begin{enumerate}[i.]
      \item Show $f$ is not 1-1.
      \item Show $f$ is not onto. 
      \item Let $A = \{-3,-2,-1,0,1, 2, 3\}$. Find $f(A) = \{ f(x) \;|\; x \in A \}$ \quad (the image of the set $A$ under the map $f$). 
      \item Let $B = \{0,1,2,3,4,5,6\}$. Find $f^{-1}(B) = \{ x \in \mathbb{Z} \;|\; f(x) \in B \}$ \quad (the inverse image of $B$). 
      \end{enumerate}

   \vspace{0.1in}

   \item Let $g: X \to Y$. Prove that $g$ is onto if and only if \quad $g^{-1}(B) \not= \phi$ (the inverse image of $B$ is non-empty) for all non-empty subsets of $Y$: \quad $\phi \not= B \subset Y$.
   
   \vspace{0.1in}
   
   Recall that for $A \subseteq X$ and $B \subseteq Y$ \dots
   
   \[ f(A) = \{ f(x) \;|\; x \in A \} \subseteq Y \qquad \mbox{and} \qquad f^{-1}(B) = \{ x \in X \;|\; f(x) \in B \} \subseteq X \]
             
   \end{enumerate}

   \vspace{0.2in}
   
   \item[\#2] Dihedral groups: generators and relations style. Recall that \dots
\[ D_n = \left\langle x,y \;|\; x^n=1, \  y^2=1, \mbox{ and } (xy)^2=1 \right\rangle = \left\{ 1,x,x^2,\dots,x^{n-1},y,xy,x^2y,\dots,x^{n-1}y \right\} \]

   \begin{enumerate}[(a)]
   \item Write down the Cayley table for $D_4$. Determine the contents of the center of $D_4$ (i.e., $Z(D_4) = \{???\}$).
   
   {\it Note:} Recall that $Z(G) = \{ a \in G \;|\; \text{for all } g\in G,\; ag=ga\}$ is the {\it center} of a group $G$.
      
   \vspace{0.1in}

   \item Fill out the following table for $D_6$:
   
   \begin{tabular}{|r||c|c|c|c|c|c|c|c|c|c|c|c|} \hline
element $g=$ & 1 & $x$ & $x^2$ & $x^3$ & $x^4$ & $x^5$ & $y$ & $xy$ & $x^2y$ & $x^3y$ & $x^4y$ & $x^5y$ \Large\strut \\ \hline
inverse $g^{-1}=$ & 1 &  &  &  &  &  &  &  &  &  &  &  \Large\strut \\ \hline
order $|g|=$ & 1 &  &  &  &  &  &  &  &  &  &  &  \Large\strut \\ \hline
   \end{tabular}

   \vspace{0.1in}
   
   \item Simplify $yx^7y^{-3}x^{11}y^8x^{-2}$ in $D_{6}$.
   
   \end{enumerate}

   \vspace{0.2in}

\item[\#3] Let $H = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \in \mathbb{Z}^{2 \times 2} \;\middle|\;a+b+c=0 \right\}$. Show $H$ is a subgroup of $\mathbb{Z}^{2 \times 2}$. 

{\it Note:} $\mathbb{Z}^{2 \times 2}$ is an Abelian group under matrix {\it addition}.

   \vspace{0.2in}

\item[\#4] Let $H$ and $K$ be subgroups of a group $G$. Show that $H \cap K$ (the intersection of $H$ and $K$) is a subgroup of $G$.

\vspace{0.2in}

\item[RESUBMIT] Type up Homework \#2 Problem \#3 and its solution in \LaTeX. 

\fbox{\parbox{6in}{
Let $a,b,n \in \mathbb{Z}$ and $n>1$. Suppose that $\mathrm{gcd}(a,n)=1$ and $\mathrm{gcd}(b,n)=1$. Show that $\mathrm{gcd}(ab,n)=1$.
}}

     When typing this problem up, write it up carefully: Restate the problem. Write in complete sentences.  

\end{description}

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