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

Azad, Azizollah; Britnell, John R. and Gill, Nick (2015). Nilpotent covers and non-nilpotent subsets of finite groups of Lie type. Forum Mathematicum, 27(6)

## Abstract

Let~ be a finite group, and~ an element of~. A subgroup~ of~ is said to be if it is nilpotent, and has nilpotency class at most~. A subset~ of~ is said to be if it contains no two elements~ and~ such that the subgroup is -nilpotent. In this paper we study the quantity~, defined to be the size of the largest non--nilpotent subset of~.

In the case that~ is a finite group of Lie type, we identify covers of~ by -nilpotent subgroups, and we use these covers to construct large non--nilpotent sets in~. We prove that for groups of fixed rank , there exist constants and such that , where is the number of maximal tori in .
%the ambient algebraic group which are stable under the Frobenius endomorphism associated with .

In the case of groups~ with twisted rank~, we provide exact formulae for~ for all . If we write for the level of the Frobenius endomorphism associated with and assume that , then these formulae may be expressed as polynomials in with coefficients in .

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