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{\bf Math 177 HW 5 Due Friday, Feb 23 at noon}
\begin{enumerate}
\setcounter{enumi}{33}
\item Recall that if $G$ and $H$ are groups, their {\it direct product} is defined as the set 
$$G \times H=\{(g, h)\,|\, g\in G, h\in H\}, $$ with the operation given by 
$$(a,b)(c,d)=(ac, bd).$$ 

When is the product of cyclic groups cyclic? 
\item\label{A4 presentation} Find a presentation of $A_4$ by taking the following steps:
\begin{enumerate}
\item Let $a=(1\,2\,3)$ and $b=(1\,2\,4)$. Show that $a$ and $b$ generate $A_4$.
\item Draw the Cayley graph $\Gamma$ of $A_4$ using the generators $a$ and $b$. This graph is planar, so make sure you draw it in the plane without any edges crossing each other.
\item Choose a maximal tree $T$ in $\Gamma$. The graph $\Gamma$ will have 24 edges and the tree $T$ will have have 11 edges. This leaves 13 edges in $\Gamma$ that are not in $T$.
\item Label each face of $\Gamma$ with the word that describes the loop given by going around the boundary of the face. (A face is a  region of the plane in the complement of $\Gamma$.) Call the set of words you get $R$.
\item For each of the 13 edges in $\Gamma$ that are not in $T$, write down the relation determined by that edge.
Show that each relation is a consequence of the relations in $R$.
\item What is the final presentation, with generators $a$ and $b$, of $A_4$?
\end{enumerate}

\item The goal of this exercise is to find a presentation for $S_4$, by building on the work that we did in Exercise~\ref{A4 presentation}.
\begin{enumerate}
\item Let $c=(1\,2)$. Show that $S_4=A_4 \cup (1\,2)A_4$.
\item Imagine taking two copies of the the graph $\Gamma$ and placing one in the plane $z=0$ in $\mathbb R^3$ and the other directly above it in the plane $z=1$. Keep the one in the plane $z=0$ labeled exactly as the Cayley graph of $A_4$. Suppose we connect each vertex in the upper graph to the vertex immediately below it in the lower graph with two edges: one labeled $(1\,2)$ going up and a second one labeled $(1\, 2)$ going down. Can you label the upper graph so that the whole thing is now the Cayley graph of $S_4$? Are the elements of the coset $(1\,2)A_4$ the vertices of the upper graph?

\item Using the Cayley graph of $S_4$ that you just constructed, choose a maximal tree $\cal T$ that consists of the tree $T$ that you chose for $A_4$ in Exercise~\ref{A4 presentation}, a copy of that tree directly above it, and one vertical edge labeled $c$ connecting the lower tree with the upper tree. (Take $c$ to connect $(1)$ and $(1\,2)$.)

\item Write down the relations that come from the faces of the lower graph, the upper graph, and every ``vertical face''. Call this set $R$.

\item Show that every relation that comes from an edge in $S_4-{\cal T}$ is a consequence of the relations in $R$.

\item What is the presentation you obtain for $S_4$?

\end{enumerate}

\item Find a presentation of $D_n$ with two generators: one a rotation and one a reflection.

\item Find a presentation of $D_n$ with two reflections as generators.

\item Let $G$ be the group defined by the presentation 
$$G=\langle x, y \,|\, xy=yx \rangle.$$
Show that $G$ is isomorphic the direct product $\mathbb Z \times \mathbb Z$. 

\item Show that the groups given by the presentations
$$\langle a, b\,|\, a^2=b^3\rangle$$ and 
$$\langle x,y\,|\, xyx=yxy \rangle$$ are isomorphic.
 

\end{enumerate}
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