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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 7\\
Due Tuesday, October 31}\end{large}\end{center}



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\begin{enumerate} 

\item Let $x \in D_{2n}$ be an element that is not a power of $r$. Use the properties of the dihedral group that we proved in class to show the following.
\begin{enumerate}
\item $x$ has order 2.
\item $rx = xr^{-1}.$
\end{enumerate}

\item Let $G$ be the group of rigid motions of a cube. (i.e. label the 8 vertices, and rotate the cube in any fashion as long as it sits in the same orientation, just as we did with the square in class.) Calculate $|G|$. Make sure you explain your answer.

\item For each of the following subsets $H$ of the given group $G$, either prove that $H$ is a subgroup of $G$ or show it's not a subgroup.

\begin{enumerate}
\item $G = \lv \R-\{0\}, \cdot \rv $ and $H = \{x \in \R-\{0\} \mid x^2 \in \Q\}.$
\item $G = \lv \R, + \rv $ and $H = \{x \in \R  \mid x^2 \in \Q\}.$
\item $G = D_{2n}$ and $H = \{ r^i s \mid i \in \{0,1, \ldots, n-1\}\} \cup \{e\}.$
\item $G = \lv  \{f:\R \to \R\}, + \rv$ and $H = \{f \in G \mid f(1)=0\}.$
\end{enumerate}

\item Let $G$ be a finite group such that $|G| = n >2.$ Show that there is no subgroup $H$ such that $|H| = n-1.$

\item Let $G$ be an abelian group. Let $H$ be the subset of $G$ consisting of all elements of finite order. Show that $H \leq G$. This is called the \textit{\textbf{torsion subgroup}} of $G$.

\item Let $n \in \Zp$, and consider the group $G = \Z \times \Z_n$. Find the torsion subgroup of $G$.

\item Let $H$ and $K$ be subgroups of $G$. Show that $H \cap K$ is a subgroup of $G$.

\item Let $A$ be an abelian group and fix $n \in \Z$. Prove that the given sets are subgroups of $A$.
\begin{enumerate}
\item $\{a^n \mid a \in A\}$
\item $\{a \in A \mid a^n =e\}$
\end{enumerate}

\item Show that if $n \geq 3$, then $\{x \in D_{2n} \mid x^2 = e\}$ is not a subgroup of $D_{2n}$. (Thus, 8.(b) is not true if the abelian assumption is removed.)


\end{enumerate}
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