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{\bf Math 177 HW 7 Due Monday, April 2 at 5 pm}
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\begin{enumerate}

\setcounter{enumi}{43}

\item Given a group $G$ and elements $x$ and $y$ in $G$, the  {\it commutator of $x$ and $y$} is $xyx^{-1}y^{-1}$.The commutator  $xyx^{-1}y^{-1}$ is sometimes denoted $[x,y]$. The {\it commutator subgroup} of $G$, denoted $[G,G]$  is the subgroup generated by  all commutators.
\begin{enumerate}
\item Show that the commutator subgroup of $G$ is normal.
\item Show that $G$ is Abelian if and only if the commutator subgroup is trivial.
\item Show that $G/[G,G]$ is Abelian.
\item Show that the commutator subgroup is the smallest normal subgroup of $G$ such that the quotient of $G$ by this subgroup is Abelian. In other words, show that if $N$ is a normal subgroup of $G$ such that $G/N$ is Abelian, then $[G,G]\subset N$.
\end{enumerate}

\item (Exercise 2, Chapter 4, Johnson) Let $G=\langle X\,|\,R\rangle$. Show that $G/[G,G]$ has presentation
$$G/[G,G]=\langle X\,|\, R, [X,X]\rangle,$$
where $[X,X]$ is the set of all relations $[x,y]=1$ for all $x, y\in X$.

\item (Exercise 3, Chapter 4, Johnson) The {\it von Dyck} group $D(l,m,n)$ is defined as 
$$D(l,m,n)=\langle x,y\,|\,x^l=1, y^m=1, (xy)^n=1 \rangle.$$ Example 5, Chapter 4 of Johnson, shows that $D(l,m,n)\cong D(n,m,l)$. Show that
$$D(l,m,n)\cong D(m,l,n)\cong D(-l,m,n)$$
and conclude that $D(l,m,n)$ is independent of the order or signs of $l,m$, and $n$.

\item (Exercise 4, Chapter 4, Johnson) What group is $D(l,m,n)$ in the following cases?
\begin{enumerate}
\item One of $l,m,n$ is 1.
\item Two of $l,m,n$ are 2.
\end{enumerate}

\item (exercise 10, Chapter 4, Johnson) Show that the group 
$$G=\langle a,b,c,d\,|\, ab=c, bc=d, cd=a, da=b\rangle$$
is cyclic and find its order.

\item (Exercise 17, Chapter 4, Johnson) Let $G=\langle x,y\,|\, x^2y=y^2x, x^8=1\rangle$. Draw a van Kampen diagram that shows that $y^8=1$ in $G$.
\end{enumerate}
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