On Groups Associated with Transformation Semigroups

For a semigroup S of transformations (total or partial) of a finite n-element set X_n, denote by G_S the group of all the permutations h of X_n that preserve S under conjugation. It is shown that, unless S contains certain nilpotents and has a very restricted form, the alternating group Alt_n may not serve as G_S, so that Alt_n ⊆ G_S implies that G_S=S_n, and S is an S_n-normal semigroup.