On groups associated with transformation semigroups (Q1806046)

Denote by \({\mathcal P}_n\) the semigroup of all partial transformations on a finite set \(X_n\). Denote by \(S_n\) the symmetric group of permutations of \(X_n\) and let \(S\) be any subsemigroup of \({\mathcal P}_n\). An automorphism \(\varphi\) of \(S\) is defined to be inner if there exists an \(h\in S_n\) such that \(\varphi(\alpha)=h\alpha h^{-1}\) for all \(\alpha\in S\) and we emphasize the fact that \(\varphi\) is induced by \(h\) by writing \(\varphi=\varphi_h\). We denote by \(\text{Inn} S\) the group of all inner automorphisms of \(S\) and we let \(G_S=\{h\in S_n:\varphi_h\in\text{Inn} S\}\). Finally, denote by \(\text{Alt}_n\) the alternating group on \(X_n\). In the main theorem, the author proves that if \(S\) is any subsemigroup of \({\mathcal P}_n\) with \(\text{Alt}_n\subseteq G_S\) and if \(n\geq 3\), \(n\neq 0\bmod 4\) and certain nilpotents satisfy an additional condition, then \(G_S=S_n\) and \(\Aut S=\text{Inn} S\).