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\begin{document}
\openup 1\jot


\large
\centerline{\rm \bf  Math 301   Homework 10 (due on November 15, 2023)}\vskip.1in

\vskip.2in
\noindent{\Large\bf Section 7.1}
\vskip.1in
\noindent{\bf    4.}  	Determine whether the set $G$ is a group under the operation $*$. \vskip.1in
 
 {\bf (a)} $G = \{2, 4, 6, 8\} $ in $\Z_{10}$;  $a * b = ab$ \vskip.1in
   \vskip.1in
  
 {\bf (b)} $G =\Z$;  $a * b = a-b$ \vskip.1in
 \vskip.1in
 
 
 {\bf (c)} 	$G = \{n\in \Z|n\mbox{ is odd }\}$; $a* b =a+ b$ \vskip.1in
 
 \vskip.2in
 
 {\bf (d)}  $G = \{2^x|x\in \Q  \}$; $a*b = ab$
\vskip.1in

{\bf   8.} Use Theorem 2.10 to list the elements of each of these groups: $U_{ 4 }$, $U_{ 6 }$, $U_{ 10 }$, $U_{ 20 }$, $U_{30  }$.     


{\bf  10.}    Show that $G =\left\{\left.\(\begin{array}{cc} a&b\\ -b&a\end{array}\)\right |a,b\in \R,\mbox{   not both  }0\right\}$  is an abelian group under matrix multiplication.  
\vskip.1in
{\bf   24.} Prove that  the set of nonzero real numbers is a group under the operation $*$ defined by \vskip.1in
\centerline{ $a*b=\left\{\begin{array}{ll}ab&\mbox{ if }a>0\\[1mm] a/b&\mbox{ if }a<0.\end{array}\right.$}  \vskip.1in

\noindent
{\bf   30.} A partial operation table for a group  $G=\{e,a,b,c,d,f\}$ is shown below. Complete the table.\vskip.1in

\centerline{$\begin{array}{c|cccccc}
& e &  a &  b &  c &  d &  f   \\
\hline
 e &e &  a &  b &  c &  d &  f   \\
 a &a &  b &e & d& & \\
 b  &b  &    & & & & \\
 c  & c & f   & & & &a \\
 d  &d  &    & & & & \\
  f & f &    & & & &  \end{array}$} 
  
\noindent{\Large\bf Section 7.2}

\vskip.1in 
 \noindent{\bf   7.} Find the order of the given element. \begin{itemize} 
\item[(b)]   $ \(\begin{array}{ccccccc} 1 & 2  & 3& 4 & 5 &6  &7 \\     2&  3 & 7 & 5 & 1 & 4 & 6\end{array}\)$ in $S_7$.
\item[(d)]     $ \(\begin{array}{rr} -\dfrac{1}{2} &  \dfrac{1}{2} \\  &\\ -\dfrac{3}{2} & -\dfrac{1}{2}\end{array}\)$ in $GL(2,\R)$.

\end{itemize} 
 
\vskip.1in 
 \noindent{\bf   12.} If $a,\ b\in G$ and $n$ is any integer. Show that $(aba^{-1})^n=ab^na^{-1}$. \vskip.1in

\noindent{\bf 20.}   
 \begin{itemize} 
\item[(a)]      Show that $a=\(\begin{array}{rr} 0&1\\ -1&-1\end{array}\)$ has order $3$ in $GL(2,\R)$ and $b=\(\begin{array}{rr} 0&-1\\ 1&0\end{array}\)$ has order $4$.  

\item[(b)]    Show that $ab$ has infinite order.   

\end{itemize} 
\vskip.1in
\noindent{\bf  24.} If  $(ab)^2=a^2b^2$ for all $a,\ b\in G$,    prove that $G$ is abelian.  \vskip.1in

 
 
\noindent {\bf  34.} Suppose $G$ has order 4, but no element of order 4.

 \begin{itemize} 

\item[(a)]     Prove that  no element of $G$ has order 3.  [{\it Hint:} If $|g|=3$, then  $G$ consists of four distinct elements $ g,g^2, g^3=e,d $     Now $gd$ must be one of these four elements. SHow that each possibility leads to contradiction.] \vskip.1in

\item[(b)]  Explain why every nonidentity element of $G$ has order 2.    \vskip.1in
\item[(c)]    Denote the elements of $G$ by $e,\ a,\ b,\ c$ and write out the operation table for $G$.  
\end{itemize} 


\end{document}



