On Groups Associated with Transformation Semigroups

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.