Imperial College London
Professor Martin Liebeck

ProfessorMartinLiebeck

Faculty of Natural Sciences, Department of Mathematics

Head of Pure Mathematics Section/Prof of Pure Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8490m.liebeck Website

 
 
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Location

 

665Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Burness:2019:10.1007/s00209-018-2101-6,
author = {Burness, T and Liebeck, MW and Shalev, A},
doi = {10.1007/s00209-018-2101-6},
journal = {Mathematische Zeitschrift},
pages = {741--760},
title = {The length and depth of algebraic groups},
url = {http://dx.doi.org/10.1007/s00209-018-2101-6},
volume = {291},
year = {2019}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups   G=G0>G1>>Gt=1 , where each   Gi  is a maximal connected subgroup of   Gi−1 . We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical   Ru(G)  and the dimension of a Borel subgroup B of the reductive quotient   G/Ru(G) . In particular, a simple algebraic group of rank r has length   dimB+r , which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds   12dimG . We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G / R(G) is bounded above in terms of the chain difference of G.
AU  - Burness,T
AU  - Liebeck,MW
AU  - Shalev,A
DO  - 10.1007/s00209-018-2101-6
EP  - 760
PY  - 2019///
SN  - 0025-5874
SP  - 741
TI  - The length and depth of algebraic groups
T2  - Mathematische Zeitschrift
UR  - http://dx.doi.org/10.1007/s00209-018-2101-6
UR  - http://hdl.handle.net/10044/1/59109
VL  - 291
ER  -
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