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\begin{document}
{\bf Math 177 HW 6 Due Friday, Mar 23 at noon}
\bigskip

Suppose that $G$ is a group generated by the subset $X$ and $R$ is any subset of $G$. Then a word $w$ in $X$ is a {\it consequence} of  $R$ if $w$ can be changed into the empty word in $X$ (the identity in $G$) by a finite sequence of the following two operations:
\begin{enumerate}
\item Insert or delete a subword of the form $x x^{-1}$ or $x^{-1} x$ where $x\in X$.
\item Insert or delete the subword $r$ or $r^{-1}$, where $r \in R$.
\end{enumerate}
 If $w_1$ and $w_2$ are words in $X$, then we say that $w_1$ and $w_2$ are {\it $R$-equivalent}, denoted $w_1 \sim_R w_2$,  if $w_1$ can be changed into $w_2$ by a finite sequence of the above two operations. Thus, a word is a consequence of $R$ if and only if it is $R$-equivalent to the identity. It is easy to show that $R$-equivalence is in fact an equivalence relation. (Every word in $X$ is equivalent to itself by doing a sequence of no operations; if we can go from $w_1$ to $w_2$ via a sequence of the allowable steps, then we can reverse the sequence to go from $w_2$ to $w_1$; if we can go from $w_1$ to $w_2$ by a sequence of allowable steps and then from $w_2$ to $w_3$ by a sequence of allowable steps, then by stringing all the steps together, we can go from $w_1$  to $w_3$.)

\begin{enumerate}

\setcounter{enumi}{38}

\item Let $G$ be a group  and let $R$ be a subset of $G$. Show that each of  the following two descriptions defines a normal subgroup of $G$ and that they are the same.
\begin{enumerate}
\item The intersection of all normal subgroups of $G$ that contain $R$.
\item The set of elements
$$\{g_1 r_1^{\epsilon_1}g_1^{-1} g_2 r_2^{\epsilon_2}g_2^{-1}\cdots g_n r_n^{\epsilon_n}g_n^{-1}\,|\,n\in \mathbb N, g_i\in G, r_i\in R, \epsilon_i \in \{-1, 1\}\}.$$
\end{enumerate}
Moreover, the normal subgroup of $G$ defined by any one of the above descriptions is the smallest normal subgroup containing $R$, meaning that it is contained in any normal subgroup containing $R$.

\item Let $G$ be a group generated by the subset $X$ and let $R$ be a subset of $G$. Let $N$ be the smallest normal subgroup containing $R$. The goal of this exercise is to show that $N$ is exactly the set of consequences of $R$.

\begin{enumerate}
\item Let $\overline R$ be the set of consequences of $R$. Show that $\overline R$ is a normal subgroup containing $R$. Thus $N\subset \overline R$ since $N$ is the smallest normal subgroup containing $R$.
\item Suppose that $w_1$ and $w_2$ are words in $X$ and that $r\in R$. Show that if $w_1w_2\in N$, then $w_1r^{\pm1}w_2 \in N$. This may be the hardest part. You need to use all of the following facts:
\begin{enumerate}
\item $wN=Nw$ for any $w$.
\item $wN=N$ if and only if  $w\in N$.
\item $w_1N=w_2N$ if and only if $w_1^{-1}w_2 \in N$.
\end{enumerate}

\item Show that if $w_1\in N$ and $w_1\sim_R w_2$, then $w_2\in N$.
\item Show that if $w\sim_R 1$, then $w\in N$.
\item Show that $\overline R \subset N$. Hence $N=\overline R$.
\end{enumerate}
 
\item Let $G$ be a group generated by the subset $X$ and let $R$ be a subset of $G$. Let $\overline R$ be the smallest normal subgroup containing $R$ and let  $w_1$ and $w_2$ be words in $X$. Prove that the following are equivalent:
\begin{enumerate}
\item The words $w_1$ and $w_2$ are in the same coset of $\overline R$.
\item The words   $w_1$ and $w_2$ are $R$-equivalent.
\item The word $w_1^{-1}w_2$ is $R$-equivalent to 1.
\end{enumerate}

\item Let $G$ be the group defined by the presentation 
$$G=\langle x, y \,|\, xyx^{-1}y^{-1}=1 \rangle.$$
This means that $G=F(\{x,y\})/\overline{R}$ where $F(\{x,y\})$ if the free group on $\{x,y\}$ and $\overline R$ is the smallest normal subgroup of $F(\{x,y\})$ containing the set  $R=\{xyx^{-1}y^{-1}\}$. The goal of this exercise is to show that the function $f:\mathbb Z \times \mathbb Z \to G$ given by $F((n,m))=x^ny^m\overline R$ is an isomorphism.
\begin{enumerate}
\item Show that if $w$ is any word in $x$ and $y$, then $w\sim_R x^ny^m$, where $n$ and $m$ are the total exponent sums of $x$ and $y$, respectively, in $w$.
\item Show that the words  $w_1$ and $w_2$ are $R$-equivalent if and only if the total exponent sum of both $x$ and $y$ is the same for both $w_1$ and $w_2$.

\item Show that $f$ is a homomorphism.
\item Show that $f$ is a bijection and hence an isomorphism.
\end{enumerate}

\item If $V$ is a finite dimensional vector space and $W$ is a subspace, then the dimension of $W$ is less than or equal to the dimension of $V$. The analog of this for free groups is NOT true: If $F$ is a free group of rank $n$ then $F$ can have a subgroup that is a free group of larger rank! The object of this exercise is to give such an example. This is exercise 15 in Chapter 1 of ``Presentations of Groups'' by Johnson.

Let $H$ be the subset of reduced words of even length in $F=F(\{x,y\})$.
\begin{enumerate}
\item Show that $H$ is is a subgroup generated by 
$$a=x^2, b=xy,c=xy^{-1}.$$
\item If $w$ is any word in $\{a, b, c\}$, then we can convert it to a word in $x$ and $y$ by replacing $a$ with $x^2$, $b$ with $xy$, and $c$ with $xy^{-1}$. There are thirty 2-letter reduced words in $\{a, b, c\}$. Show that after converting each to a word in $\{x, y\}$, and reducing if necessary, the resulting reduced words each have length at least 2.
\item More generally, show that any reduced word of length $\ell$ in $\{a, b, c\}$, when converted to a word in $\{x, y\}$, will result in a reduced word of at least length $\ell$.
\item Use Proposition 3 in Chapter 1 of Johnson to conclude that $H$ is a free group of rank 3.
\end{enumerate}

 

\end{enumerate}
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