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

Azizollah Azad, John R. Britnell, Nick Gill

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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 $ $ 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 $D_r N \leq \omega_\infty(L) \leq E_r N$, 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 $\{0,1\}$.
Original languageEnglish
Pages (from-to)3745-3782
Number of pages34
JournalForum Mathematicum
DOIs
Publication statusPublished - 6 Nov 2015

Keywords

  • math.GR
  • 20D60, 20E07, 20G40

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