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{\bf Math 177 HW 3 Due Friday, Feb 9 at noon}
\begin{enumerate}
\setcounter{enumi}{17}

\item Let $G$ be a group and $S$ a subset of $G$. Let $H$ be the set of all elements of $G$ than can be expressed as products of elements in $S$ and their inverses. 
\begin{enumerate}
\item Show that $H$ is a subgroup of $G$ and that $H$ is generated by $S$. 
\item Show that $H$ is the smallest subgroup of $G$ that contains $S$. That is, if $H'$ is any subgroup of $G$ that contains $S$, then $H$ is contained in $H'$.
\end{enumerate}

\item Suppose that $G$ is a group and $H_1$ and $H_2$ are subgroups of $G$. Show that $H_1 \cap H_2$ is a subgroup of $G$. Give an example of a group $G$ that contains two subgroups whose union is not a subgroup.

\item Prove or disprove this statement: Every subgroup of a cyclic group is cyclic.

\item Let $\mathbb Z$ be the group of integers under addition. What are all subgroups of $\mathbb Z$?

\item Let $n>1$ be given. Find all subgroups of $C_n$, the cyclic group of order $n$. (Warning: The answer depends on whether $n$ is prime or not.)

\item Given any group $G$, the {\it center} of $G$, denoted $Z(G)$, is defined as all elements of $G$ that commute with every element of $G$. That is, 
$$Z(G)=\{g \in G\,|\, gh=hg \mbox{ for all } h\in G\}.$$
\begin{enumerate}
\item Show that $Z(G)$ is a subgroup of $G$.
\item Show that $Z(G)=G$ if and only if $G$ is Abelian.
\item Show that $Z(G)$ is always a normal subgroup of $G$.
\end{enumerate}

\item Let $n>1$ be given. Let $r$ and $f$ be the isometries of the regular $n$-gon that are counter-clockwise rotation through an angle of $2 \pi/n$, and reflection though a line containing the center of the $n$-gon and a vertex of the $n$-gon, respectively. We may now write all the elements of $D_n$ in terms in $r$ and $f$ as follows:
$$D_n=\{1, r, r^2, \dots, r^{n-1}, f, fr, \dots, fr^{n-1}\}.$$ 
\begin{enumerate}
\item Show that $r^k f=f r^{n-k}$ for all $0<k<n$.
\item Assume that $n$ is an odd prime. Find all subgroups of $D_n$. (Hint: Consider all subgroups generated by the subset $S$, where $S$ has one element, or two elements, or three elements, etc. For example, what if $S$ has two elements, one of which is a rotation, $r^k$, and one of which is a flip, $fr^j$?)

\item Continuing to assume that $n$ is an odd prime, what is the center $Z(D_n)$?

\item Find all subgroups of $D_4$. Four is not prime, so things are a bit different.

\item What is the center of $D_4$?
\end{enumerate}


\item Show that any subgroup $H$ of a group $G$ that has index two is normal. (The {\it index} of $H$ in $G$ is the number of right cosets of $H$ in $G$, which is always the same as the number of left cosets of $H$ in $G$.)
This is Exercise 52 in our book and the solution is in the back of the book! Try to do it first yourself. Although in our book, the result is only stated in the case where $G$ is finite.
\end{enumerate}
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