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\begin{document}
\title{Weekly Homework 3}
\author{Michael Mayer\\
Math 4377: Algebraic Structures}
\maketitle
\begin{problem}{1}
\text{ }\\
Find all Solutions to the equation $x^{2} \oplus x = [0] $ in $\Z_{4}$
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{2}
\text{ }\\
$(1)$ Prove: If $[a] \in \Z_{n}$ is a unit, then $[a]$ is not a zero divisor.\\
$(2)$ Prove: If $[b] \in \Z_{n}$ is a zero divisor, then $[b]$ is not a unit.
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{3}
\text{ }\\
Show that every nonzero element of $\Z_{n}$ is either a unit or a zero divisor.
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{4}
\text{ }\\
Suppose that $[a]$ is a unit in $\Z_{n}$ and $[b]$ is an element of $\Z_{n}$. Prove that the equation $[a]x = b$ has exactly one solution in $\Z_{n}$
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{5}
\text{ }\\
Suppose that $[a]$ and $[b]$ are both units in $\Z_{n}$. Show that the product $[a] \cdot [b]$ is also a unit in $\Z_{n}.$ (Note that this confirms closure under multiplication in the group $U_{n})$.
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{6}
\text{ }\\
Which of the following are Groups? Which of the following are not groups, and why?\\
\indent (1) $G = \{{2, 4, 6, 8}\}$ in $\Z_{10}$. Where $a \star b = ab$\\
\indent (2) $G = \Q^{\ast}$, where $a \star b = \frac{a}{b}$\\
\indent (3) $G = \Z$, where $a \star b = a - b$\\
\indent (4) $G = \{ {2^{x}\mid x \in \Q} \}$, where $a \star b = ab$\\
\end{problem}
\begin{proof}
\end{proof}
\begin{problem}{7}
\text{ }\\
Consider the set $Q =$ \{ $\pm$1, $\pm$i, $\pm$j, $\pm$k\} of the complex matrices as follows:\\
\[
1=
\begin{bmatrix}
1 & 0\\
0 & 1\\
\end{bmatrix}
\]
\[
i=
\begin{bmatrix}
i & 0\\
0 & $-$i\\
\end{bmatrix}
\]
\[
j=
\begin{bmatrix}
0 & 1\\
$-$1 & 0\\
\end{bmatrix}
\]
\[
k=
\begin{bmatrix}
0 & i\\
i & 0\\
\end{bmatrix}
\]
Show that $Q$ is a group under matrix multiplication by writing out its multiplicaiton table. (Note: $Q$ is called the quartenion group).
\end{problem}
\begin{proof}
\end{proof}
\end{document}
