  We consider the permutation group algebra defined by Cameron and show that if
the permutation group has no finite orbits, then no homogeneous element of
degree one is a zero-divisor of the algebra. We proceed to make a conjecture
which would show that the algebra is an integral domain if, in addition, the
group is oligomorphic. We go on to show that this conjecture is true in certain
special cases, including those of the form H Wr S and H Wr A, and show that in
the oligormorphic case, the algebras corresponding to these special groups are
polynomial algebras. In the H Wr A case, the algebra is related to the shuffle
algebra of free Lie algebra theory. We finish by considering some integer
sequences which arise from certain of these groups.
