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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 1.2  \\ Spring 2023 \hfill Due January 18}}}

\large

\vspace{1cm}

\begin{exercise}
Given $n \in \N$ and $x \in \Z$, let $R_n(x)$ denote the remainder when $x$ is divided by $n$ using the Division Algorithm.   

\begin{itemize}

\item[\bf a.]  Prove that for any $a,b \in \Z$, $R_n(a) = R_n(b)$ if and only if $a-b$ is divisible by $n$ in $\Z$, i.e.\ there is a $c \in \Z$ so that $a-b = cn$.

\item[\bf b.]  Prove that for any $a, b \in \Z$ one has $R_n(a + b) = R_n(R_n(a)+b)$ and $R_n(ab) = R_n(R_n(a)b)$.

\item[\bf c.]  Recall that for $a, b \in \Z$ we defined $a \oplus b = R_n(a+b)$ and $a \otimes b = R_n(ab)$.  Show that $\oplus$ and $\otimes$ are both associative.  [{\em Suggestion.}  Given $a,b,c \in \Z$, consider $R_n(a+b+c)$ and $R_n(abc)$.]

\end{itemize}

\end{exercise}

\begin{exercise}
Let
\[
G = \left\{ \pm \mat{1}{0}{0}{1}, \pm \mat{i}{0}{0}{-i}, \pm \mat{0}{-1}{1}{0} ,\pm \mat{0}{i}{i}{0} \right\}.
\]
\begin{itemize}

\item[\bf a.]  Show that $G$ is closed under matrix multiplication.  

\item[\bf b.]  Prove that $G$ is a group under matrix multiplication.  Is it abelian?

\end{itemize}
\end{exercise}

\begin{exercise}
Let $G$ be a group and $e_0 \in G$.  Given $x, y \in G$, define a new binary operation $\ast$ on $G$ by 
\[
x \ast y = x e_0^{-1} y.
\]  
Prove that $G$ is a group under $\ast$.
\end{exercise}

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