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dc.contributor.authorAndré, Jorge
dc.contributor.authorAraúo, Joāo
dc.contributor.authorCameron, Peter Jephson
dc.identifier.citationAndré , J , Araúo , J & Cameron , P J 2016 , ' The classification of partition homogeneous groups with applications to semigroup theory ' , Journal of Algebra , vol. 452 , pp. 288-310 .
dc.identifier.otherPURE: 240185816
dc.identifier.otherPURE UUID: ecd73140-f5ee-4407-9f6c-e357696ccf50
dc.identifier.otherScopus: 84957927053
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055757
dc.identifier.otherWOS: 000372391900016
dc.description.abstractLet λ=(λ1,λ2,...) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω:={1,...,n}. An ordered partition P=(A1,A2,...) of Ω has type λ if |Ai|=λi.Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P=(A1,A2,...) and Q=(B1,B2,...) of Ω of type λ, there exists g ∈ G with Aig=Bi for all i. A group G is said to be λ-homogeneous if, given two ordered partitions P and Q as above, inducing the sets P'={A1,A2,...} and Q'={B1,B2,...}, there exists g ∈ G such that P'g=Q'. Clearly a λ-transitive group is λ-homogeneous.The first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H<=Sn. Given a non-invertible transformation a in Tn-Sn and a group G<=Sn, we say that (a,G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, {a,G}\G= {a,H}\H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.8). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5). This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.
dc.relation.ispartofJournal of Algebraen
dc.rights© 2016, Elsevier, Inc. All rights reserved. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at
dc.subjectTransformation semigroupsen
dc.subjectPermutation groupsen
dc.subjectPrimitive groupsen
dc.subjectQA Mathematicsen
dc.titleThe classification of partition homogeneous groups with applications to semigroup theoryen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Statisticsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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