arXiv subjects

Azizollah Azad

Publications and source records attributed to Azizollah Azad.

3 recordsLinked to original sources

arXiv2025

Non-commuting graph of AC-groups: as matroids

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph {\Gamma}G (called non-commuting graph of G) as follows: Take G\Z(G) as the vertices of {\Gamma}G and join x and y, whenever $xy \not= yx$. In this paper, we show that a finite group G is an AC-group, if and only if, the associated non-commuting graph of G is a matroid. Leveraging the properties of matroids, we further delve into the characteristics of AC-groups. Additionally, we provide a formula to compute the clique number of the non-commuting graph of AC-groups, offering a new perspective on the structure of these groups

math.GR↗

arXiv2022

Upper Bounds For The Diameter Of A Direct Power Of Solvable Groups

Let G be a finite group with a generating set A. By the (symmetric) diameter of G with respect to A we mean the maximum over g in G of the length of the shortest word in (A union A inverse)A expressing g.By the (symmetric) diameter of G we mean the maximum of (symmetric) diameter over all generating sets of G. Let n greater than or equal to 1, by G power n we mean the n-th direct power of G. For n greater than or equal to 1 and finite non-abelian solvable group G we find an upper bound, growing polynomially with respect to n, for the symmetric diameter and the diameter of G power n.

math.GR↗

arXiv2014

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

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 $ω_c(G)$, 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 $D_r N \leq ω_\infty(L) \leq E_r N$, 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 $ω_c(L)$ 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 $ω_\infty(G)$ may be expressed as a polynomial in $q$ with coefficients in $\{0,1\}$.

math.GR↗