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

Abstract

Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty \}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $<x,y>$ is $c$-nilpotent. In this paper we study the quantity ${\omega }_c(G)$, 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 $L$. We prove that for groups $L$ of fixed rank $r$, there exist constants $D_r$ and $E_r$ such that $D_{r}N\le {\omega }_{\infty }(L)\le E_{r}N$, 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 ${\omega }_c(L)$ for all $c\in \mathbb{Z}\cup \{\infty \}$. If we write $q$ for the level of the Frobenius endomorphism associated with $L$ and assume that $q>5$, then ${\omega }_{\infty }(G)$ may be expressed as a polynomial in $q$ with coefficients in $\{0,1\}$.

This resubmission provides a corrected statement of Theorem 5.5, and makes a number of other small corrections and improvements. The paper will appear in Forum Mathematicum. 33 pages