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Azizolah Azad

Azizolah Azad

Academic rank: Associate Professor
ORCID: https://orcid.org/0000-0002-7950-0977
Education: PhD.
ScopusId: 14420904600
Faculty: Science
Address: Arak University
Phone:

Research

Title
Nilpotent covers and non-nilpotent subsets of finite groups of Lie type
Type
JournalPaper
Keywords
. Non-nilpotent set, nilpotent subgroup, nilpotent cover, regular semisimple element, regular unipotent element, finite simple group of Lie type.
Year
2015
Journal Forum Mathematicum
DOI
Researchers Azizolah Azad ، John R Britnell ، Nick Gill

Abstract

. Let G be a finite group and c be an element of ZC [ ¹1º. A subgroup H of G is said to be c-nilpotent if it is nilpotent and has nilpotency class at most c. A subset X of G is said to be non-c-nilpotent if it contains no two elements x and y such that the subgroup hx; yi is c-nilpotent. In this paper we study the quantity !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 the group L. We prove that for groups L of fixed rank r , there exist constants Dr and Er such that Dr N !1 .L/ Er 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 !c .L/ for all c 2 ZC [ ¹1º. If we write q for the level of the Frobenius endomorphism associated with L and assume that q > 5, then !1 .L/ may be expressed as a polynomial in q with coefficients in ¹0; 1º.