On the Automorphism Group of the Centralizer of an Idempotent in the Full Transformation Monoid

摘要

Let e be an idempotent in the monoid T(X) of all functions from a set X into itself. Let C(e) be the centralizer of e in T(X). It has recently been shown that the unit and automorphism groups of C(e) are canonically isomorphic. Our goal is to furnish an alternative proof of this fact and make the observation that automorphism group of C(e) is isomorphic to the direct product Π_(i∈I)(Sym(A_i)Sym(B_i)) of wreath products of symmetric groups, where the sets I, A_i, B_i are defined in terms of e.