Nilpotent covers and non-nilpotent subsets of finite groups of Lie type
Nilpotent covers and non-nilpotent subsets of finite groups of Lie type
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李型有限群的幂零覆盖和非幂零子集
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发表时间：
2013
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通讯作者：
Nick Gill
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作者：
A. Azad;John R. Britnell;Nick Gill
设$G$是有限群，$c$是$mathbb{Z}cup {infty}$的元素. G$的子群H$称为{it $c$-幂零}，如果它是幂零的，且幂零类至多为$c$。G$的子集X$称为{it non $c$-幂零}，如果它不包含两个元素x$和y$使得子群$ $是$c$-幂零的。本文研究了Ω_c（G），它定义为L的最大非c-幂零子集的大小。
在有限Lie群的情形下，我们确定了L的c-幂零子群覆盖，并利用这些覆盖构造了L中的非c-幂零大集.本文证明了对于固定秩r的群L，存在常数D_r和E_r，使得D_r N leq omega_infty（L）leq E_r N，其中N是L中极大环面的个数.
在扭秩为1的群L的情形下，我们给出了ω_c（L）的精确公式。如果我们将$q$写为与$L$相关联的Frobenius自同态的水平，并且假设$q>5$，则$omega_infty（G）$可以表示为在$q$中的多项式，其系数在${0，1}$中。
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 $cinmathbb{Z}cup {infty}$. 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 ${0,1}$.