Groups and actions in transformation semigroups

Let be a transformation semigroup of degree . To each element we associate a permutation group acting on the image of , and we find a natural generating set for this group. It turns out that the -class of is a disjoint union of certain sets, each having size equal to the size of . As a consequence, we show that two -classes containing elements with equal images have the same size, even if they do not belong to the same -class. By a certain duality process we associate to another permutation group on the image of , and prove analogous results for the -class of . Finally we prove that the Schützenberger group of the -class of is isomorphic to the intersection of and . The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper. Received 16 December 1996; in final form 18 February 1997