Permutation Groups, a Related Algebra and a Conjecture of Cameron

Abstract

Abstract. 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.

Galois groups of tamely ramified p-extensions

“…There are algebras arising in other areas of mathematics whose graded pieces have the same dimensions, namely (i) the free Lie algebra generated by one generator in degree 1 and one in degree 2 (arising in work on multizeta values and quantum field theory [13]) and (ii) Cameron's permutation group algebra [19] of C 2 A, where A is the group of all order-preserving permutations of the rationals.…”