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\begin{flushright}
  Your Name Here
\end{flushright}


\begin{center}
  \textbf{Math 71}\\ Homework 2
\end{center}



\begin{itemize}


\item[p27 \#1c]Compute the order of each of the elements in $D_{10}$.
 

\item[p33. \#1] Let $\sigma$ be the permutation
  \[1\mapsto 3\quad 2\mapsto 4\quad 3\mapsto 5\quad 4\mapsto 2\quad
    5\mapsto 1,
  \]
  and let $\tau$ be the permutation
  \[1\mapsto 5\quad 2\mapsto 3\quad 3\mapsto 2\quad 4\mapsto 4\quad
    5\mapsto 1.
  \]
  Find the cycle decompositions of each of the following permutations:
  $\sigma, \tau, \sigma^2, \sigma\tau, \tau\sigma, \tau^2\sigma$.

 

\item[p33. \#4a] Compute the order of each of the elements in $S_3$.
  
\item[p33. \#10] Prove that if $\sigma$ is the $m$-cycle
  $(a_1\,a_2\, \dots \,a_m)$, then for all $i$ with $1 \le i \le m$,
  we have $\sigma^i(a_k) = a_{k+i}$ where $k+i$ is read modulo $m$
  (least positive residue). Deduce that $|\sigma| = m$.


\item[p33. \#15] Prove that the order of an element in $S_n$ equals
  the least common multiple of the lengths of the cycles in its cycle
  decomposition.

  
\item[p35. \#4] Show that if $n>1$ is not prime, then $\Z/n\Z$ is not
  a field.

  
\item[p35. \#5]Show that $GL_n(F)$ is a finite group if and only if
  $F$ has only a finite number of elements.
  

\item[p35. \#8] Show that $GL_n(F)$ is nonabelian for any $n\ge 2$ and
  any $F$.


\item [p40. \#9] Prove that $D_{24}$ and $S_4$ are not isomorphic.

 
\item [p40. \#13] Let $G$ and $H$ be groups, and $\varphi:G \to H$ a
  homomorphism.  Prove that the image of $\varphi$, $\varphi(G)$, is a
  subgroup of $H$.  Prove that if $\varphi$ is injective, then
  $G \cong \varphi(G)$.

  
\item [p40. \#14] Let $G$ and $H$ be groups, and $\varphi:G \to H$ a
  homomorphism. Define the \textbf{kernel} of $\varphi$ to be
  $\ker(\varphi) := \{g \in G\mid \varphi(g) = 1_H\}$, the fiber over
  the identity of $H$.  Prove that $\ker(\varphi)$ is a subgroup of
  $G$ and that $\varphi$ is injective if and only if
  $\ker(\varphi) = \{1_G\}$.

  
\end{itemize}


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