[1] The Schützenberger groups associated with different H-classes are distinct, but the groups associated with two different H-classes contained in the same D-class of a semigroup are isomorphic.
In fact, there are two Schützenberger groups associated with a given H-class, with each being antiisomorphic to the other.
[4] Let S be a semigroup and let S1 be the semigroup obtained by adjoining an identity element 1 to S (if S already has an identity element, then S1 = S).
Let H be an H-class of the semigroup S. Let T(H) be the set of all elements t in S1 such that Ht is a subset of H itself.
Each t in T(H) defines a transformation, denoted by γt, of H by mapping h in H to ht in H. The set of all these transformations of H, denoted by Γ(H), is a group under composition of mappings (taking functions as right operators).