On the automorphism group of the centralizer of an idempotent in the full transformation monoid.
\(T(X)\) denotes the semigroup, under composition, of all selfmaps of a nonempty set \(X\). Let \(e\) be an idempotent in \(T(X)\) and let \(C(e)\) be the centralizer of \(e\) in \(T(X)\). It is known that the group of units of \(C(e)\) and the automorphism group of \(C(e)\) are isomorphic. The author gives another proof of this fact and, in addition, he shows that the automorphism group of \(C(e)\) is isomorphic to the direct product of wreath products of symmetric groups. To be specific, define an equivalence relation \(\sim\) on \(X\) by \(\sim=\{(x,y)\in X\times X:e(x)= e(y)\}\). Denote by \(C_x\) the \(\sim\)-equivalence class of \(X\) containing \(x\). Define an equivalence relation \(\simeq\) on \(e(X)\) by \(\simeq=\{(x,y)\in e(X)\times e(X):|C_x|=|C_y|\}\) and denote by \(E_x\) the \(\simeq\)-equivalence class of \(e(X)\) containing \(x\). Finally, denote the wreath product of two groups \(G\) and \(H\) by \(G\wr H\). Let \(I\) be a collection of representatives for \(\simeq\) in \(e(X)\) and let \(\text{Sym}(X)\) denote the symmetric group on the set \(X\). In the concluding result of the paper, the author proves that the automorphism group of \(C(e)\) is isomorphic to \(\prod_{i\in I}(\text{Sym}(C_i\setminus\{i\})\wr\text{Sym}(E_i))\).
scientific article; zbMATH DE number 2173796