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\section*{Math 3140: Homework 5}


\begin{enumerate}
\item[A.] 
\begin{enumerate}
\item[7.2] Verify that the set of integers $\{1,3,7,9,11,13,17,19\}$ form a group under multiplication modulo 20.  Explain why this group is not isomorphic to $\ZZ_8$.
\item[7.5]  Show that the function 
$$\begin{array}{r@{\ }ccc}\varphi: & G & \longrightarrow & G\\ & g & \mapsto & g^{-1}\end{array}$$
is an isomorphism if and only if $G$ is abelian.
 \item[7.9]  Suppose $G$ is cyclic with generator $x\in G$.  Show that if $\varphi:G\rightarrow H$ is an isomorphism, then $\varphi$ is completely determined by $\varphi(x)$.  Show that $H=\langle\varphi(x)\rangle$.
\end{enumerate}
\item[B.]  
\begin{enumerate}
\item[8.7]  Using Cayley's Theorem, explicitly find an isomorphic copy of $D_3$ inside $S_6$. 
\item[8.10] For each $w\in S_n$, let $\tilde{w}\in S_{2n}$ be the permutation given by
$$\tilde{w}(j)=\left\{\begin{array}{ll} w(j), & \text{if $1\leq j\leq n$,}\\ w(j-n)+n, & \text{if $n+1\leq j\leq 2n$.}\end{array}\right. $$
\begin{enumerate}
\item[(a)] Describe the relationship between the cycle decomposition of $w$ and the cycle decomposition of $\tilde{w}$.
\item[(b)] Show that the function that sends $w\mapsto \tilde{w}$ is an isomorphism between $S_n$ and a subgroup of $A_{2n}$. 
\end{enumerate}
\item[8.11]  Let $G$ be the full symmetry group of a regular tetrahedron $T$, and adopt the notation of Figure 7.2 in the book.  Find the symmetry $q$ of $T$ which induces the transposition $(1,2)$ of the vertices, and show that $qr$ induces the $4$-cycle $(1,2,3,4)$.  Check that $qr$ is neither a rotation nor a reflection, but is the product of three reflections.  Count the symmetries of $T$ and prove that $G$ is isomorphic to $S_4$.
\end{enumerate}
\end{enumerate}

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