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\begin{document}
\openup 1\jot


\large
\centerline{\rm \bf  Math 301   Homework 12  (due on December 6, 2023)}\vskip.1in

\vskip.2in
\noindent{\Large\bf Section 7.5}
\vskip.1in
\noindent{\bf    3, 4.}    Express each permutation    as a product of disjoint cycles {\bf and} as a product of transpositions.
\vskip.1in
      \begin{itemize} 
\item[(b)]   $ \(\begin{array}{ccccccccc} 1 & 2  & 3& 4 & 5 &6  &7&8&9 \\     3&  5 & 1 & 2 & 4 & 6& 8 & 9 & 7\end{array}\)$ \vskip.1in
\item[(d)]   $(14)(27)(523)(34)(1472)$
\end{itemize} 
\vskip.1in

\noindent{\bf  6.}     Find the order of each permutation. 
\vskip.1in
  \begin{itemize} 
\item[(b)]      $(123)(456)$.
\item[(c)]  $(123)(435)$.
  \item[(d)]     $ (1234)(4231)$.   
\item[(e)]  $(1234)(24)(43215)$.
\end{itemize} 
\vskip.1in
 \noindent{\bf   14} Show that $\beta=(1236)(5910)(465)(5678)$ has order 21 in $S_n\ (n\ge 10)$.
\vskip.1in

\noindent{\bf   21.}    Find  the order of $\sigma^{1000}$, where $\sigma$ is the permutation   \vskip.2in
 \centerline{$\(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\
 3&7&8&9&4&5&2&1&6\end{array}\)$.}
\vskip.1in



  \vskip.1in 
\noindent{\bf   22.}   Show that $S_{10}$ contains elements of orders 10, 20, 30. Does it contain an element of order 40? \vskip.1in
  
\noindent{\Large\bf Section 8.1}

\vskip.1in 
\vskip.1in
\noindent{\bf    6.} Let $K=\langle 3\rangle$; $G=U_{32}$.  List the distinct $K$ cosets of $K$ in $G$.  \vskip.1in  
\vskip.1in
      


\noindent{\bf  20.}   A group has fewer than 100 elements and subgroups of order 10 and 25.  What is the order of the group?   \vskip.1in
\vskip.1in
\noindent{\bf   26.} Prove that a group of order 8 must contain an element of order $2$.  \vskip.1in
  
\noindent{\bf   29.}  Let $H$ and $K$ be subgroups of a finite group $G$ such that $K\subseteq H$, $[G:H]$ is finite, and $[H:K]$  is finite Prove that $  [G:K]= [G:H][H:K]$.  [{\it Hint:} Lagrange.]\vskip.1in


\noindent{\bf    33.}    
\vskip.1in
      \begin{itemize} 
\item[(a)]  If $a$ and $b$ each have order 3 in a group and $a^2=b^2$, prove that $a=b$.   [{\it Hint:} What are $a^{-1}$ and $b^{-1}$.]\vskip.1in
\item[(b)]   If $G$ is a finite group, prove that there is an even number of elements of order 3 in $G$.
\end{itemize}

\end{document}



