The length and depth of compact Lie groups

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Title: The length and depth of compact Lie groups
Authors: Burness, TC
Liebeck, MW
Shalev, A
Item Type: Journal Article
Abstract: Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . In this paper, we introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G. We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on dimG′ in terms of the chain difference of G, which is its length minus its depth.
Issue Date: 16-May-2019
Date of Acceptance: 9-Apr-2019
URI: http://hdl.handle.net/10044/1/70284
DOI: https://dx.doi.org/10.1007/s00209-019-02324-7
ISSN: 0025-5874
Publisher: Springer (part of Springer Nature)
Journal / Book Title: Mathematische Zeitschrift
Copyright Statement: © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: 0101 Pure Mathematics
General Mathematics
Publication Status: Published online
Online Publication Date: 2019-05-16
Appears in Collections:Pure Mathematics
Mathematics



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