Putnam 1976 B2

B-2. Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_{1}} B^{n_{2}} A^{n_{1}} \cdots B^{n_{k}}$, where $n_{1}, \ldots, n_{k}$ are any integers, and $A^{0}=B^{n}=1$ as usual. Also, suppose that $A^{4}=B^{7}=A B A^{-1} B=1, A^{2} \neq 1$, and $B \neq 1$. (a) How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$ ? (b) Write each such square as a word in $A$ and $B$.