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A. Abdollahi

Publications and source records attributed to A. Abdollahi.

18 recordsLinked to original sources
arXiv2026

Regular sets of circulant quartic graphs

For a graph $Γ=(V,E)$ and nonnegative integers $a$ and $b$, a nonempty proper subset $C \subset V$ is called an $(a,b)$-regular set if every vertex in $C$ has exactly $a$ neighbors in $C$, and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In this paper, we study the existence of such sets in connected Cayley graph $Γ= \operatorname{Cay}(\mathbb{Z}_n, S)$. We establish a necessary and sufficient condition for the existence of $(0, |S|)$-regular sets and identify additional conditions under which no such set can exist. We further prove that $(|S|, 0)$-regular sets do not occur in $Γ$, and more generally, that no connected Cayley graph $\operatorname{Cay}(G,S)$ contains a $(1, |S|)$-regular set. As a main result, we determine the existence and nonexistence of $(a,b)$-regular sets in connected circulant quartic graphs for all possible values of $a$ and $b$.

math.CO↗
arXiv2026

The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors

The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on \(n\) letters under the Hamming metric, this problem is closely related to the parameter $N(n,r)$, the maximum intersection size of two Hamming balls of radius $r$. While previous work has resolved \(N(n,r)\) for small radii (\(r \leq 4\)) and established asymptotic bounds for larger \(r\), we present new exact formulas for \(r \in \{5,6,7\}\) using group action techniques. In addition, we develop a formula for \(N(n,r)\) based on the irreducible characters of the symmetric group \(S_n\), along with an algorithm that enables computation of \(N(n,r)\) for larger parameters, including cases such as \(N(43,8)\) and \(N(24,14)\).

math.GR↗
arXiv2022

New Bounds on the Size of Permutation Codes With Minimum Kendall $τ$-distance of Three

We study $P(n,3)$, the size of the largest subset of the set of all permutations $S_n$ with minimum Kendall $τ$-distance $3$. Using a combination of group theory and integer programming, we reduced the upper bound of $P(p,3)$ from $(p-1)!-1$ to $(p-1)!-\lceil\frac{p}{3}\rceil+2\leq (p-1)!-2$ for all primes $p\geq 11$. In special cases where $n$ is equal to $6,7,11,13,14,15$ and $17$ we reduced the upper bound of $P(n,3)$ by $3,3,9,11,1,1$ and $4$, respectively.

math.CO↗
arXiv2020

A conjecture of Cameron and Kiyota on sharp characters with prescribed values

Let $ χ$ be a virtual (generalized) character of a finite group $ G $ and $ L=L(χ)$ be the image of $ χ$ on $ G-\lbrace 1 \rbrace $. The pair $ (G, χ) $ is said to be sharp of type $ L $ if $|G|=\prod _{ l \in L} (χ(1) - l) $. If the principal character of $G$ is not an irreducible constituent of $χ$, the pair $(G,χ)$ is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in $1988$. This conjecture states that if $(G,χ)$ is sharp and $|L|\geq 2$, then the inner product $(χ,χ)_G$ is uniquely determined by $ L $. We then prove that this conjecture is true in the case that $(G,χ) $ is normalized, $χ$ is a character of $ G $, and $ L $ contains at least an irrational value.

math.RT↗
arXiv2017

Non-abelian finite groups whose character sums are invariant but are not Cayley isomorphism

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^α=T$ for some automorphism $α$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_ν^S=M_ν^T$ for all positive integers $ν$, where $M_ν^S$ denotes the set $\big\{\sum_{s\in S}χ(s) | χ(1)=ν, χ\text{ is a complex irreducible character of } G \big\}$. It was asked by László Babai [\textit{J. Combin. Theory Ser. B}, {\bf 27} (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [\textit{Comm. Algebra}, {\bf 43} (12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However it is known that there are finite abelian non CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders $20$ and $42$ which are not CI-groups. We also list all BI-groups of orders up to $30$.

math.GR↗
arXiv2013

A note on noninner automorphisms of order $p$ for finite $p$-groups of coclass 2

In this note, the existence of noninner automorphisms of order 2 for finite 2-groups of coclass 2 is proved. Combining our result with a recent one due to Y. Guerboussa and M. Reguiat (see arXiv:1301.0085), we prove that every finite $p$-group of coclass 2 has a noninner automorphism of order $p$ leaving the center elementwise fixed.

math.GR↗
arXiv2012

Finite 2-groups of Class 2 with Specific Automorphism Group

In this paper we classify all finite 2-groups of class 2 for which every automorphism of order 2 leaving the Frattini subgroup elementwise fixed is inner. We prove that every such group G is isomorphic to Q(n; r) = for some positive integers r; n such that 2 < 2r <= n; and every automorphism of Q(n; r) of order 2 leaving the Frattini subgroup elementwise fixed is inner.

math.GR↗
arXiv2011

Commutativity pattern of finite non-abelian $p$-groups determine their orders

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $Γ_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\setminus Z(G)$ as the vertices of $Γ_G$ and join two distinct vertices $x$ and $y$, whenever $xy\neq yx$. Here, we prove that "the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups"; this means that if $P$ is a finite non-abelian $p$-group such that $Γ_P\cong Γ_H$ for some group $H$, then $|P|=|H|$.

math.GR↗
arXiv2011

G-frame representation and Invertibility of g-Bessel Multipliers

In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-frame) is a controlled frame (resp. weighted frame).

math.FA↗
arXiv2010

Right 4-Engel elements of a group

We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $ ^G$ is nilpotent of class at most 7 for each right 4-Engel elements $x$ of $G$.

math.GR↗
arXiv2009

When right n-Engel elements of a group form a subgroup?

Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $ ^G$ is nilpotent of class at most 7 for each $x\in R_4(G)$. Using a group constructed by Newman and Nickel we also show that, for each $n\geq 5$, there exists a nilpotent group of class $n+2$ containing a right $n$-Engel element $x$ and an element $a\in G$ such that both $[x^{-1},_n a]$ and $[x^{k},_n a]$ are of infinite order for all integers $k\geq 2$. We finish the paper by proving that at least one of the following happens: (1) There is an infinite finitely generated $k$-Engel group of exponent $n$ for some positive integer $k$ and some 2-power number $n$. (2) There is a group generated by finitely many bounded left Engel elements which is not an Engel group.

math.GR↗
arXiv2009

On the right and left 4-Engel elements

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $ $ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left 4-Engel elements and $b$ is an arbitrary element of $G$. Furthermore we prove that for any prime $p$ and any element $a$ of finite $p$-power order in a group $G$ such that $a^{\pm 1}\in L_4(G)$, $a^4$, if $p=2$, and $a^p$, if $p$ is an odd prime number, is in the Baer radical of $G$.

math.GR↗
arXiv2009

On the clique number of non-commuting graphs of certain groups

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple graph $Γ$ the maximum size of a complete subgraph of $Γ$ is called the clique number of $Γ$ and it is denoted by $ω(Γ)$. In this paper we characterize all non-solvable groups $G$ with $ω(\mathcal{A}_G)\leq 57$, where the number 57 is the clique number of the non-commuting graph of the projective special linear group $\mathrm{PSL}(2,7)$. We also complete the determination of $ω(\mathcal{A}_G)$ for all finite minimal simple groups.

math.GR↗
arXiv2008

Configuration of nilpotent groups and isomorphism

The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.

math.GR↗