Abstract
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 ${\Pi}_{i\in I}( \Sym(A_i)\wr \Sym(B_i))$ of wreath products of symmetric groups, where the sets $I$, $A_i$, $B_i$ are defined in terms of $e$.
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Szechtman, F. On the Automorphism Group of the Centralizer of an Idempotent in the Full Transformation Monoid. Semigroup Forum 70, 238–242 (2005). https://doi.org/10.1007/s00233-004-0141-1
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DOI: https://doi.org/10.1007/s00233-004-0141-1