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Orbit coherence in permutation groups

Lookup NU author(s): Dr John Britnell

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Abstract

For G ≤ Sym.(Δ), let π.(G) be the set of partitions of Δ given by the cycles of elements of G. Under the refinement order, π(G) admits join and meet operations. We say that G is join- or meet-coherent if π(G) is join- or meet-closed, respectively. The centralizer in Sym. (Δ) of any permutation g is shown to be meet-coherent, and joincoherent subject to a finiteness condition. Hence if G is a centralizer in Sn, then π(G) is a lattice. We prove that wreath products, acting imprimitively, inherit join-coherence from their factors. In particular automorphism groups of locally finite, spherically homogeneous trees are join-coherent. We classify primitive join-coherent groups of finite degree, and also join-coherent subgroups of Sn normalizing an n-cycle. We show that if π(G) is a chain, then there is a prime p such that G acts regularly on each of its orbits as a subgroup of the Prufer p-group, with G being isomorphic to an inverse limit of these subgroups. © de Gruyter 2014.

Publication metadata

Author(s): Britnell JR, Wildon M

Publication type: Article

Publication status: Published

Journal: Journal of Group Theory

Year: 2014

Volume: 17

Issue: 1

Pages: 73-109

Print publication date: 01/01/2014

Online publication date: 10/01/2014

Acceptance date: 01/01/1900

ISSN (print): 1433-5883

ISSN (electronic): 1435-4446

Publisher: Walter de Gruyter and Co.

URL: https://doi.org/https://doi.org/10.1515/jgt-2013-0058

DOI: 10.1515/jgt-2013-0029

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