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)

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

Abstract

Let~$G$ be a finite group, and~$c$ an element of~$\longintegers$. 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 $\langle x,y\rangle$ is $c$-nilpotent. In this paper we study the quantity~$\omegac{G}$, defined to be the size of the largest non-$c$-nilpotent subset of~$G$.

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 $D_r N \leq \omega_\infty(L) \leq E_r N$, where $N$ is the number of maximal tori in $L$.
%the ambient algebraic group which are stable under the Frobenius endomorphism associated with $L$.

In the case of groups~$L$ with twisted rank~$1$, we provide exact formulae for~$\omegac{L}$ for all $c\in\longintegers$. If we write $q$ for the level of the Frobenius endomorphism associated with $L$ and assume that $q>5$, then these formulae may be expressed as polynomials in $q$ with coefficients in $\mathbb{Z}[\frac12]$.

Viewing alternatives

Download history

Metrics

Public Attention

Altmetrics from Altmetric

Number of Citations

Citations from Dimensions

Item Actions

Export

About

Recommendations