Weekly Homework 3
Forfatter:
Mike Mayer
Sidst opdateret:
10 år siden
Licens:
Creative Commons CC BY 4.0
Resumé:
Week 3 homework for Dr Martin's class
\begin
Opdag hvorfor 18 millioner mennesker verden rundt stoler på Overleaf med deres arbejde.
\begin
Opdag hvorfor 18 millioner mennesker verden rundt stoler på Overleaf med deres arbejde.
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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}