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Nilpotent covers and non-nilpotent subsets of finite groups of Lie type

Lookup NU author(s): Dr John Britnell

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Abstract

© by De Gruyter 2015.Let G be a finite group and c be an element of Z+ {∞}. A subgroup H of G is said to be c-nilpotent if it is nilpotent and has nilpotency class at most c. A subset X of G is said to be non-c-nilpotent if it contains no two elements x and y such that the subgroup hx; yi is c-nilpotent. In this paper we study the quantity ωc(G), defined to be the size of the largest non-c-nilpotent subset of L. In the case that L is a finite group of Lie type, we identify covers of L by c-nilpotent subgroups, and we use these covers to construct large non-c-nilpotent sets in the group L. We prove that for groups L of fixed rank r, there exist constants Dr and Er such that DrN ≤ ω∞(L) ≤ ErN, where N is the number of maximal tori in L. In the case of groups L with twisted rank 1, we provide exact formulae for ωc(L) for all c ∈ Z+ {∞}. If we write q for the level of the Frobenius endomorphism associated with L and assume that q > 5, then ω∞(L) may be expressed as a polynomial in q with coefficients in {0; 1}.

Publication metadata

Author(s): Azad A, Britnell JR, Gill N

Publication type: Article

Publication status: Published

Journal: Forum Mathematicum

Year: 2015

Volume: 27

Issue: 6

Pages: 3745-3782

Print publication date: 01/11/2015

Online publication date: 06/11/2015

Acceptance date: 02/04/2014

ISSN (print): 0933-7741

ISSN (electronic): 1435-5337

Publisher: Walter de Gruyter GmbH

URL: https://doi.org/10.1515/forum-2013-0176

DOI: 10.1515/forum-2013-0176

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