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\title{Algebra Homework Set 4}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 5 October. Homework must be typeset and submitted by email as a PDF file. 

\begin{enumerate}


\item Let $G$ be the free group on $n$ generators, say $\{ x_1, \ldots, x_n \}$. 
  Define three groups as follows:

  \begin{enumerate}

  \item $G_1 = G/[G, G]$.

  \item $G_2$ is the group freely generated by $X$ subject to the relations $x_i x_j = x_j x_i$, that
    is $G_2 = G/N$ where $N$ is the least normal subgroup containing the words $x_i X_j x^{-1} x_j^{-1}$.

  \item $G_3 = {\mathbb Z}^n$.

  \end{enumerate}

  Prove that the groups $G_1, G_2, G_3$ are isomorphic. Hint: They are all ``the most general abelian
  group with $n$ generators''.

\item  Let $G = \langle a, b : b a b^{-1} = a^{-1} \rangle$. Prove that $G$ is infinite.

\item  Let $F_2$ be the free group on two generators. Prove that every finite symmetric group $S_n$ 
  is a homomorphic image of $F_2$. Is {\em every} finite group a homomorphic image of $F_2$?
  
\item Let $G$ be the free group on $n$ generators. Let $H = G/[G, G]$ and let $H_1 = \{ h^2 : h \in H \}$.
  Compute $[H: H_1]$, and use your answer to conclude that all free generating sets for $G$ have size $n$. 
  Not for credit exercise for those who know a bit of set theory: Replace $n$ by
  an infinite cardinal $\kappa$. 

\item Let $F$ be a group and let $X$ be a set with $X \subseteq F$. Prove that the following are equivalent:

  \begin{itemize}

  \item $F$ is freely generated by $X$. 

  \item For every set $A$ and every function $f$ from $X$ to $\Sigma_A$,
    there is a unique action of $F$ on $A$ such that $x \cdot a = f(x)(a)$
    for all $x$ and $a$. 

  \end{itemize}
  
  
  
\end{enumerate}

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