The structure of the automorphism group of a Croisot-Teissier semigroup (Q2639972)

The authors complete their investigation of the automorphisms of the Croisot-Teissier semigroups. Two papers precede it and I. Levi also contributed to the first of these papers. Here, they are finally able to completely describe the automorphism groups of the Croisot-Teissier semigroups. Let S be a Croisot-Teissier semigroup and let \(\omega\) be the congruence on S consisting of all pairs (f,g) such that \(hfk=hgk\) for all h,k\(\in S\). Then the map T from Aut S onto Aut S/\(\omega\) defined by \(T(\phi)[f]=[\phi (f)]\) is evidentally a homomorphism and \(\phi\in \ker T\) if and only if (f,\(\phi\) (f))\(\in \omega\). Ker T is denoted by \(\omega\)-Aut S and is referred to as the group of \(\omega\)-stabilizing automorphisms of S. In the main theorem they show that Aut S is the split extension of \(\omega\)-Aut S by CAut S where CAut S is a certain group of automorphisms of S which is isomorphic to a subgroup, which they previously described, of the automorphism group of a Croisot-Teissier semigroup of elementary type. Moreover, \(\omega\)-Aut S is, itself, the split extension of \(\eta\)-Aut S by \(\mu\)-RPAut \(S\times \nu\)-RPAut S, where both \(\eta\)-Aut S and \(\mu\)-RPAut S are direct products of full symmetric groups, while \(\nu\)-RPAut S is a direct product of wreath products of full symmetric groups. In conclusion, it is appropriate to point out that the information in this paper, together with that in the two papers mentioned earlier, adds considerably to what was previously known about the Croisot-Teissier semigroups.