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\title{\textbf{Homework 3}}

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\normalsize
\section*{Section 2.4}
\textbf{Exercise 28.} \vskip1ex
Let $p \geq 2$. Prove that if $2^p-1$ is prime then $p$ must also be prime.
\vskip1ex
\emph{Proof:}
\hfill $\square$

\vskip2ex
\textbf{Exercise 31.} \vskip1ex
Using the fact that $2$ is prime, show that there do not exist integers $p$ and $q$ such that $p^2 = 2q^2$, and therefore, $\sqrt{2}$ cannot be a rational number.
\vskip1ex
\emph{Proof:}
\hfill $\square$
\vskip2ex
\section*{Section 3.5}
\textbf{Exercise 1 (b, d).}
\begin{enumerate}
\item[(b)] Find $x$ satisfying $5x+1 \equiv 13 \ (\text{mod} \ 23)$.
\vskip1ex
\emph{Solution: } 
\item[(d)] Find $x$ satisfying $9x \equiv 3 \ (\text{mod} \ 5)$.
\vskip1ex
\emph{Solution: }
\end{enumerate}
\hfill $\square$
\vskip2ex

\textbf{Exercise 2(c, d).}  \vskip1ex
Which of the following multiplication tables defined on the set $G = \{a,b,c,d\}$ form a group? Support your answer in each case.
\begin{enumerate}
\item[(c)]
  \noindent\begin{tabular}{c | c c c c}
    $\circ$ & a & b & c & d  \\
    \cline{1-5}
    a & a & b & c & d  \\
    b & b & c & d & a  \\
    c & c & d & a & b  \\
    d & d & a & b & c  \\
  \end{tabular}
\vskip1ex
\emph{Solution: } 
\item[(d)]
  \noindent\begin{tabular}{c | c c c c}
    $\circ$ & a & b & c & d  \\
    \cline{1-5}
    a & a & b & c & d  \\
    b & b & a & c & d  \\
    c & c & b & a & d  \\
    d & d & d & b & c  \\
  \end{tabular}
\vskip1ex
\emph{Solution: }
\end{enumerate}
\vskip2ex

\textbf{Exercise 6.} \vskip1ex
Give a multiplication table for the group $U(12)$.
\vskip2ex
\emph{Solution:} \vskip1ex
\hfill $\square$
\vskip2ex

\textbf{Exercise 7.} \vskip1ex
On the set $S = \R \setminus \{-1 \}$, define the following binary operation for $a, b \in S$: $$a \ast b = a+b+ab.$$
Prove that $(S, \ast)$ is an abelian group.
\vskip2ex
\emph{Proof:} \vskip1ex
\hfill $\square$
\vskip2ex

\vskip2ex
\textbf{Exercise 8.} \vskip1ex
Give an example of two elements $A$ and $B$ in $GL_2(\R)$ with $AB \neq BA$.
\vskip1ex
\emph{Solution: } \vskip1ex
\hfill $\square$
\vskip2ex

\textbf{Exercise 10.} \vskip1ex
Prove that the set of matrices of the form $\Bigg ( \begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} \Bigg ) $ is a group under matrix multiplication
(called the Heisenberg group).
\vskip1ex
\emph{Proof: } \vskip1ex
\hfill $\square$
\vskip2ex

\textbf{Exercise 14.} \vskip1ex
Given the groups $\R^* = \R \setminus {0}$ and $\Z$, let $G = \R^* \times \Z$. Define a binary operation $\circ$ on $G$ by $(a,m)\circ(b,n) = (ab,m+n)$. 
Show that $G$ is a group under this operation.
\vskip1ex
\emph{Proof: } \vskip1ex
\hfill $\square$
\vskip2ex

\textbf{Exercise 15.} \vskip1ex
Prove or disprove that every group containing six elements is abelian. 
\vskip1ex
\emph{Solution: } \vskip1ex
\hfill $\square$

\textbf{Exercise 26.} \vskip1ex
Let $U(n)$ be the group of units in $\Z_n$. If $n>2$, prove that there is an element $k \in U(n)$ such that $k^2 = 1$ and $k \neq 1$.
\vskip1ex
\emph{Solution: } \vskip1ex
\hfill $\square$



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