-element set X n has only inner automorphisms is investigated. For a semigroup S of transformations of X n, let G s denote the group of all permutations of X n that preserve S under conjugation. Fix a group G of permutations of X n and let T be a semigroup of transformations of X n maximal with respect to the property G T = G. For certain groups G we give a complete description of all such T and prove that Aut T + Inn T ≅ G.
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