On the automorphism group of the centralizer of an idempotent in the full transformation monoid.
type: Item
instance of: scholarly article
title: On the automorphism group of the centralizer of an idempotent in the full transformation monoid.
zbMATH Open document ID: 1080.20057
author: Fernando Szechtman
published in: Semigroup Forum
publication date: 2005-06-06T00:00:00Z
review text: \(T(X)\) 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 \(\prod_{i\in I}(\text{Sym}(C_i\setminus\{i\})\wr\text{Sym}(E_i))\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 2173796
zbMATH Keywords: full transformation semigroups
zbMATH Keywords: centralizers
zbMATH Keywords: idempotents
zbMATH Keywords: automorphism groups
zbMATH Keywords: wreath products
zbMATH Keywords: groups of units
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
full work available at URL: s00233-004-0141-1
OpenAlex ID: W2089464979
DOI: 10.1007/S00233-004-0141-1
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description: scientific article; zbMATH DE number 2173796