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\begin{document}
\openup 1\jot


\large
\centerline{\rm \bf  Math 301   Homework 11 (due on November 29, 2023)}\vskip.1in

\vskip.2in
\noindent{\Large\bf Section 7.3}
\vskip.1in
\noindent{\bf    6.}   Describe $\langle 2\rangle$ in the multiplicative group of nonzero elements in $\Z_{11}$.
\vskip.1in
      


\noindent{\bf  12.}      Show that the additive group $\Z_2\times \Z_4$ is not cyclic but generated by two elements. 
\vskip.1in
\noindent{\bf   14.}   Let $H$ and $ K$ be subgroups of a group $G$.    \vskip.1in
\begin{itemize} 
\item[(a)]   Show by example that $H\cup K$ need not be a subgroup of $G$.  \vskip.1in
\item[(b)]    Prove that $H\cup K$ is     a subgroup of $G$ if and only if $H\subseteq K$ or $K\subseteq H$.

\end{itemize} 
\noindent{\bf   28.} Let $G$ be an abelian group and $n$ a fixed positive integer.   \vskip.1in
\begin{itemize} 
\item[(a)]   Prove that $H=\{a\in G|a^n=e\}$ is a subgroup of $G$.  \vskip.1in
\item[(b)]    Show by example that part (a) may be false if $G$ is nonabelian.

\end{itemize}  
  
\noindent{\Large\bf Section 7.4}

\vskip.1in 
\vskip.1in
\noindent{\bf    6.}  Prove that the function $h:\Z_8\to \Z_8$ defined by $h(x)=2x$ is a homomorphism that is neither injective nor surjective.   \vskip.1in  
\vskip.1in
      


\noindent{\bf  12.}    Prove that the function $h:\R \to GL(2,\R)$ defined by $h(x)= \(\begin{array}{cc}1&0\\ x&1\end{array}\)$ is a injective homomorphism.     \vskip.1in
\vskip.1in
\noindent{\bf   14.}   Prove that the additive group $\Z_6$ is isomorphic to the multiplicative group of nonzero elements in $\Z_7$.  \vskip.1in
  
\noindent{\bf   24.} Let $G$ be a multiplicative group. Let $G^{op}$ be the set $G$ equipped with a new operation $*$ defined by $a*b=ba$. \vskip.1in
\begin{itemize} 
\item[(a)]   Prove that  $G^{op}$  is a group.
\item[(b)]    Prove that  $G\cong G^{op}$. [{\it Hint: } Corollary 7.6 may be helpful.]

\end{itemize}  
  


\end{document}



