Groups associated with finite transformation semigroups
type: Item
instance of: scholarly article
title: Groups associated with finite transformation semigroups
zbMATH Open document ID: 0966.20031
author: Donald B. McAlister
author: Inessa Levi
author: Robert B. McFadden
published in: Semigroup Forum
publication date: 2001-07-16T00:00:00Z
review text: Let \(S\) be a semigroup of mappings (perhaps partial) on \(N\), a finite \(n\)-element set, and let \(G_S\) be the group of all permutations \(h\) on \(N\) such that \(hah^{-1}\in S\) for all \(a\in S\). It is known that if \(S\) consists of total mappings and \(A_n\subseteq G_S\) then \(G_S=S_n\), the full symmetric group. This is also true under other circumstances such as \(n\) not being divisible by 4. However, the main result of this paper shows that if \(n=2m\), \(m\) even, and \(u\) is a so-called \(x\)-nilpotent of rank \(m\) then the semigroup generated by the conjugates of \(u\) with respect to the alternating group \(A_n\) is strictly contained in that generated by arbitrary conjugates of \(u\). They go on to determine the semigroups \(S\) of partial mappings such that \(G_S=A_n\), find generating sets for such \(S\), and all the ideals, automorphisms and congruences on \(S\). Finally, they address and partly answer the question as to when the semigroup generated by a map \(a\) and its conjugates with respect to a group \(G\) is equal to the semigroup generated by \(G\cup\{a\}\) with \(G\) deleted.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1545256
zbMATH Keywords: inner automorphisms
zbMATH Keywords: \(S_n\)-normal semigroups
zbMATH Keywords: semigroups of mappings
zbMATH Keywords: total mappings
zbMATH Keywords: symmetric groups
zbMATH Keywords: conjugates
zbMATH Keywords: alternating groups
zbMATH Keywords: partial mappings
zbMATH Keywords: generating sets
zbMATH Keywords: ideals
zbMATH Keywords: congruences
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1545256