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Permutation groups, a related algebra and a conjecture of Cameron. (English) Zbl 1080.20002

Author’s summary: 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 \(H\text{\,Wr\,}S\) and \(H\text{\,Wr\,}A\), and show that in the oligomorphic case, the algebras corresponding to these special groups are polynomial algebras. In the \(H\text{\,Wr\,}A\) case, the algebra is related to the shuffle algebra of free Lie algebra theory.

MSC:

20B07 General theory for infinite permutation groups
05D10 Ramsey theory
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