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The classification of partition homogeneous groups with applications to semigroup theory
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dc.contributor.author | André, Jorge | |
dc.contributor.author | Araúo, Joāo | |
dc.contributor.author | Cameron, Peter Jephson | |
dc.date.accessioned | 2017-02-05T00:32:35Z | |
dc.date.available | 2017-02-05T00:32:35Z | |
dc.date.issued | 2016-04-15 | |
dc.identifier | 240185816 | |
dc.identifier | ecd73140-f5ee-4407-9f6c-e357696ccf50 | |
dc.identifier | 84957927053 | |
dc.identifier | 000372391900016 | |
dc.identifier.citation | André , 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 . https://doi.org/10.1016/j.jalgebra.2015.12.025 | en |
dc.identifier.issn | 0021-8693 | |
dc.identifier.other | ORCID: /0000-0003-3130-9505/work/58055757 | |
dc.identifier.uri | https://hdl.handle.net/10023/10228 | |
dc.description.abstract | Let 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.format.extent | 359749 | |
dc.language.iso | eng | |
dc.relation.ispartof | Journal of Algebra | en |
dc.subject | Transformation semigroups | en |
dc.subject | Permutation groups | en |
dc.subject | Primitive groups | en |
dc.subject | Lambda-transitive | en |
dc.subject | GAP | en |
dc.subject | QA Mathematics | en |
dc.subject | T-NDAS | en |
dc.subject.lcc | QA | en |
dc.title | The classification of partition homogeneous groups with applications to semigroup theory | en |
dc.type | Journal article | en |
dc.contributor.institution | University of St Andrews. Pure Mathematics | en |
dc.contributor.institution | University of St Andrews. Statistics | en |
dc.contributor.institution | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | https://doi.org/10.1016/j.jalgebra.2015.12.025 | |
dc.description.status | Peer reviewed | en |
dc.date.embargoedUntil | 2017-02-04 |
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