Imperial College London
Professor Martin Liebeck

ProfessorMartinLiebeck

Faculty of Natural Sciences, Department of Mathematics

Head of Pure Mathematics Section/Prof of Pure Mathematics
 
 
 
//

Contact

 

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

 
 
//

Location

 

665Huxley BuildingSouth Kensington Campus

//

Summary

 

Publications

Citation

BibTex format

@article{Burness:2019:10.1112/plms.12273,
author = {Burness, T and Liebeck, M and Shalev, A},
doi = {10.1112/plms.12273},
journal = {Proceedings of the London Mathematical Society},
pages = {1464--1492},
title = {On the length and depth of  finite groups},
url = {http://dx.doi.org/10.1112/plms.12273},
volume = {119},
year = {2019}
}
Download

RIS format (EndNote, RefMan)

TY  - JOUR
AB  - An unrefinable chain of a finite group      is a chain of subgroups   =  0>  1>>    =1 , where each        is a maximal subgroup of      −1 . The length (respectively, depth) of      is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth 3. Here, we go much further by determining the finite groups of depth 3 and 4. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most 9, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a numbertheoretic result of HeathBrown, which implies that there are infinitely many nonabelian simple groups of length at most 9.Finally, we study the chain difference of      (namely the length minus the depth). We obtain results on groups with chain differences 1 and 2, including a complete classification of the simple groups with chain difference 2, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of    /  (  )  in terms of the chain difference of     , where    (  )  is the soluble radical of     .
AU  - Burness,T
AU  - Liebeck,M
AU  - Shalev,A
DO  - 10.1112/plms.12273
EP  - 1492
PY  - 2019///
SN  - 1460-244X
SP  - 1464
TI  - On the length and depth of  finite groups
T2  - Proceedings of the London Mathematical Society
UR  - http://dx.doi.org/10.1112/plms.12273
UR  - http://hdl.handle.net/10044/1/70636
VL  - 119
ER  -
Download