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\begin{document}
\title{Homework $\#$2}
\author{Due 10/2/13 at start of class }
\maketitle
\subsection*{Problem 1 (D$\&$F 1.2.2,3)} Consider the dihedral group $D_{2n}$ with the usual presentation $D_{2n}=\langle r,s\vert r^n=s^2=1, rs=sr^{-1}\rangle$. Use these generators and relations to show:
\begin{itemize}
\item[(a)] If $x\in D_{2n}$ is not a power of $r$, then $rx=xr^{-1}$.
\item[(b)] Every element of $D_{2n}$ which is not a power of $r$ has order $2$.
\end{itemize}  

\subsection*{Problem 2 (D$\&$F 1.2.13)} Recall that a regular icosahedron is a 20-faced polyhedron, with each face an equilateral triangle. Let $G$ be the group of rigid motions in $\mathbb{R}^3$ (symmetries) of a regular icosahedron. Show that $\vert G\vert=60$. 


\subsection*{Problem 3 (D$\&$F 1.3.1)} Let $\sigma$ be the permutation 
$$\begin{array}{ccccc}
1\mapsto 3 & 2\mapsto 4 & 3\mapsto 5 & 4\mapsto 2 & 5\mapsto 1
\end{array}$$
and let $\tau$ be the permutation
$$\begin{array}{ccccc}
1\mapsto 5 & 2\mapsto 3 & 3\mapsto 2 & 4\mapsto 4 & 5\mapsto 1
\end{array}$$
Find the cycle decompositions of each of the following permutations: $$\sigma, \tau, \sigma^2,\sigma\tau, \tau\sigma, \tau^2\sigma.$$

\subsection*{Problem 4 (D$\&$F 1.3.10)} Prove that if $\sigma$ is the m-cycle $(a_1 \;a_2\;\dots \;a_m)$ then for all $1\leq i\leq m$, $\sigma^i(a_k)=a_{k+i}$, where $k+i$ is replaced by its least positive residue $\Mod m$. Deduce that $\vert \sigma\vert =m$.


\subsection*{Problem 5 (D$\&$F 1.3.9,11)}
\begin{itemize}
\item[(a)] Let $\sigma$ be the 10-cycle $(1\;2\;3\;4\;5\;6\;7\;8\;9\;10)$. For which positive integers $i$ is $\sigma^i$ also a 10-cycle?
\item[(b)]Let $\tau$ be the 8-cycle $(1\;2\;3\;4\;5\;6\;7\;8)$. For which positive integers $i$ is $\tau^i$ also a 8-cycle?
\item[(c)] Let $\sigma$ be the $m$-cycle $(1\;2\;\dots\;m)$. Using parts (a) and (b), complete the following statement:
$$\sigma^i \;\text{is an m-cycle if and only if...}$$

Challenge: Prove this.
\end{itemize}

\subsection*{Problem 6 (D$\&$F 1.3.15)} 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.

\noindent [Hint: Use Problem 5 from Homework$\#$1]

\subsection*{Problem 7 (D$\&$F 1.4.4)} Show that if $n$ is not prime then $\mathbb{Z}/n\mathbb{Z}$ is not a field.

\subsection*{Problem 8 (D$\&$F 1.6.3)} Let $G,H$ be groups. If $\phi: G\rightarrow H$ is an isomorphism, prove that $G$ is abelian
if and only if $H$ is abelian. 

\subsection*{Problem 9 (D$\&$F 1.6.11)} Let $G,H$ be groups. Prove that $G\times H\cong H\times G$.

\subsection*{Problem 10 (D$\&$F 1.6.14)} Let $G,H$ be groups and $\phi: G\rightarrow H$ a homomorphism. Define the \textit{kernel} of $\phi$ to be $$\ker(\phi)=\{g\in G\vert\; \phi(g)=e_H\}.$$ Prove that $\phi$ is injective if and only if $\ker(\phi)=\{e_G\}$.

\subsection*{Problem 11 (D$\&$F 1.6.17)} Let $G$ be a group. Prove that the map from $G$ to itself defined by $g\mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.

\subsection*{Challenge Problem}
Consider the symmetric group $S_n$. Show that the two elements $(1\; 2)$ and $(1\;2\;\dots\;n)$ form a generating set for $S_n$. 
\end{document}