Putnam 1977 B6

Problem B-6 Let $H$ be a subgroup with $h$ elements in a group $G$. Suppose that $G$ has an element $a$ such that for all $x$ in $H$, $(x a)^{3}=1$, the identity. In $G$, let $P$ be the subset of all products $x_{1} a x_{2} a \cdots x_{n} a$, with $n$ a positive integer and the $x_{i}$ in H. (a) Show that $P$ is a finite set. (b) Show that, in fact, $P$ has no more than $3 h^{2}$ elements.