arXiv subjects

A. R. Moghaddamfar

Publications and source records attributed to A. R. Moghaddamfar.

18 recordsLinked to original sources

arXiv2026

Some Remarks on Super $M_{p}$-groups

Let $G$ be a finite group and $p$ be a prime divisor of $|G|$. An irreducible $p$-Brauer character $φ$ of $G$ is called super-monomial if every primitive $p$-Brauer character inducing $φ$ is linear. The group $G$ is said to be a super $M_{p}$-group if every irreducible $p$-Brauer character of $G$ is super-monomial. In this note, we investigate the conditions under which a finite group $G$ qualifies as a super $M_{p}$-group. We demonstrate that every normal subgroup of a super $M_{p}$-group of odd order is an $M_{p}$-group.

math.GR↗

arXiv2023

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups

The spectrum of a group is the set of orders of its elements. Finite groups with the same spectra as the direct squares of the finite simple groups with abelian Sylow 2-subgroups are considered. It is proved that the direct square $J_1\times J_1$ of the sporadic Janko group $J_1$ and the direct squares ${^2}G_2(q)\times{^2}G_2(q)$ of the simple small Ree groups ${^2}G_2(q)$ are uniquely characterized by their spectra in the class of finite groups, while for the direct square $PSL_2(q)\times PSL_2(q)$ of a 2-dimensional simple linear group $PSL_2(q)$, there are always infinitely many groups (even solvable groups) with the same spectra.

math.GR↗

arXiv2020

Groups that have a Partition by Commuting Subsets

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions -- with exact calculations in some cases. Finally, we examine how the size of partitions with the minimum number of parts behaves with respect to the direct product.

math.GR↗

arXiv2020

The Solubility Graph Associated With a Finite Group

Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $Γ_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements $x$ and $y$ if and only if $\langle x, y\rangle$ is a soluble subgroup of $G$. In this paper, we examine some properties of solubility graphs.

math.GR↗

arXiv2018

Simple Groups Whose Prime Graph or Solvable Graph is Split

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.

math.GR↗

arXiv2018

OD-Characterization of Some Simple Unitary Groups

The degree pattern of a finite group is the degree sequence of its prime graph in ascending order of vertices. We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise nonisomorphic finite groups with the same order and degree pattern as the group under consideration. In this article the problem of OD-characterization is solved for some simple unitary groups. It was shown, in particular, that the simple unitary groups $U_3(q)$ and $U_4(q)$ are OD-characterizable, where $q$ is a prime power $<10^2$.

math.GR↗

arXiv2018

The Complexity of Power Graphs Associated With Finite Groups

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of a clique--replaced graph and study some applications. Second, we derive some explicit formulas concerning the complexity $κ(\mathcal{P}(G))$ for various groups $G$ such as the cyclic group of order $n$, the simple groups $L_2(q)$, the extra--special $p$--groups of order $p^3$, the Frobenius groups, etc.

math.GR↗

arXiv2018

Splitting via Noncommutativity

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such that $|B_i| > 1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n = 1, 2, 3$. Finally, we show that for a nonabelian group $G$ having a strict $n$-split decomposition, the index $|G:A|$ is bounded by some function of $n$.

math.GR↗

arXiv2017

Some Infinite Matrices Whose Leading Principal Minors Are Well-known Sequences

There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.

math.NT↗

arXiv2017

The number of spanning trees of power graphs associated with specific groups and some applications

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $\langle x\rangle\subseteq \langle y\rangle$ or $\langle y\rangle\subseteq \langle x\rangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group $G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we show that the alternating group $\mathbb{A}_5$ is uniquely determined by tree-number of its power graph among all finite simple groups.

math.GR↗

arXiv2015

Several Quantitative Characterizations of Some Specific Groups

Let $G$ be a finite group and let $π(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is $π(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_ip_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)=\left(d_G(p_1), d_G(p_2), \ldots, d_G(p_k)\right)$ is said to be the degree pattern of $G$. Moreover, if $ω\subseteq π(G)$ is the vertex set of a connected component of ${\rm GK}(G)$, then the largest $ω$-number which divides $|G|$, is said to be an order component of ${\rm GK}(G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.

math.GR↗

arXiv2013

Certain properties of the power graph associated with a finite group

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some properties of the associated graph. The power graph $\mathcal{P}(G)$ of a group $G$ is a simple graph whose vertex-set is $G$ and two vertices $x$ and $y$ in $G$ are adjacent if and only if $y=x^m$ or $x=y^m$ for some positive integer $m$. We also pay attention to the subgraph $\mathcal{P}^\ast(G)$ of $\mathcal{P}(G)$ which is obtained by deleting the vertex 1 (the identity element of $G$). In the present paper, we first investigate some properties of the power graph $\mathcal{P}(G)$ and the subgraph $\mathcal{P}^\ast(G)$. We next prove that many of finite groups such as finite simple groups, symmetric groups and the automorphism groups of sporadic simple groups can be uniquely determined by their power graphs among all finite groups. We have also determined up to isomorphism the structure of any finite group $G$ such that the graph $\mathcal{P}^\ast(G)$ is a strongly regular graph, a bipartite graph, a planar graph or an Eulerian graph. Finally, we obtained some infinite families of finite groups such that the graph $\mathcal{P}^\ast(G)$ containing some cut-edges.

math.GR↗

arXiv2013

On Recognition by Order and Degree Pattern of Finite Simple Groups

Let ${\rm GK}(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that $|H|=|G|$ and $D(H)=D(G)$. A 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this paper is threefold. First, it provides the reader with a few useful and efficient tools on OD-characterizability of finite groups. Second, it lists a number of such simple groups that have been already investigated. Third, it shows that the simple groups $L_6(3)$ and $U_4(5)$ are OD-characterizable, too.

math.GR↗

arXiv2013

OD-Characterization of Some Linear Groups Over Binary Field and Their Automorphism

The Gruenberg-Kegel graph ${\rm GK}(G)=(V_G, E_G)$ of a finite group $G$ is a simple graph with vertex set $V_G=π(G)$, the set of all primes dividing the order of $G$, and such that two distinct vertices $p$ and $q$ are joined by an edge, $\{p, q\}\in E_G$, if $G$ contains an element of order $pq$. The degree ${\rm deg}_G(p)$ of a vertex $p\in V_G$ is the number of edges incident on $p$. In the case when $π(G)=\{p_1, p_2,..., p_h\}$ with $p_1< p_2< ... < p_h$, we consider the $h$-tuple $D(G)=({\rm deg}_G(p_1), {\rm deg}_G(p_2),..., {\rm deg}_G(p_h))$, which is called the degree pattern of $G$. The group $G$ is called $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ satisfying condition $(|H|, D(H))=(|G|, D(G))$. Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we first find the degree pattern of the projevtive special linear groups over binary field $L_n(2)$ and among other results we prove that the simple groups $L_{10}(2)$ and $L_{11}(2)$ are OD-characterizable (Theorem \ref{10-11}). It is also shown that automorphism groups ${\rm Aut}(L_p(2))$ and ${\rm Aut}(L_{p+1}(2))$, where $2^p-1$ is a Mersenne prime, are OD-characterizable (Theorem \ref{auto}).

math.GR↗

arXiv2013

Some Quantitative Characterizations of Certain Symplectic Groups

Given a finite group $G$, denote by ${\rm D}(G)$ the degree pattern of $G$ and by ${\rm OC}(G)$ the set of all order components of $G$. Denote by $h_{\rm OD}(G)$ (resp. $h_{\rm OC}(G)$) the number of isomorphism classes of finite groups $H$ satisfying conditions $|H|=|G|$ and ${\rm D}(H)={\rm D}(G)$ (resp. ${\rm OC}(H)={\rm OC}(G)$). A finite group $G$ is called OD-characterizable (resp. OC-characterizable) if $h_{\rm OD}(G)=1$ (resp. $h_{\rm OC}(G)=1$). Let $C=C_p(2)$ be a symplectic group over binary field, for which $2^p-1>7$ is a Mersenne prime. The aim of this article is to prove that $h_{\rm OD}(C)=1=h_{\rm OC}(C)$.

math.GR↗

arXiv2013

OD-Characterization of Certain Four Dimensional Linear Groups with Related Results Concerning Degree Patterns

The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there exists an element of order $pq$ in $G$. Let $p_1<p_2<...<p_k$ be all prime divisors of $|G|$. Then the degree pattern of $G$ is defined as ${\rm D}(G)=(deg_G(p_1), deg_G(p_2),..., deg_G(p_k))$, where $deg_G(p)$ signifies the degree of the vertex $p$ in ${\rm GK}(G)$. A finite group $H$ is said to be OD-characterizable if $G\cong H$ for every finite group $G$ such that $|G|=|H|$ and ${\rm D}(G)={\rm D}(H)$. The purpose of this article is threefold. First, it finds sharp upper and lower bounds on $\vartheta(G)$, the sum of degrees of all vertices in ${\rm GK}(G)$, for any finite group $G$ (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic $p$ of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups $L_4(19)$, $L_4(23)$, $L_4(27)$, $L_4(29)$, $L_4(31)$, $L_4(32)$ and $L_4(37)$ are OD-characterizable (Theorem 4.2).

math.GR↗

arXiv2009

Generalized Pascal Triangles and Toeplitz Matrices

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Toeplitz matrix. This equality allows us to evaluate a few determinants of generalized Pascal matrices associated to certain sequences. In particular, we obtain families of quasi-Pascal matrices whose principal minors generate any arbitrary linear subsequences F(nr+s) or L(nr+s), (n=1, 2, 3, ...) of Fibonacci or Lucas sequence.

math.RA↗

arXiv2005

The Number of Finite Groups Whose Element Orders is Given

The spectrum $ω(G)$ of a finite group $G$ is the set of element orders of $G$. If $Ω$ is a non-empty subset of the set of natural numbers, $h(Ω)$ stands for the number of isomorphism classes of finite groups $G$ with $ω(G)=Ω$ and put $h(G)=h(ω(G))$. We say that $G$ is recognizable (by spectrum $ω(G)$) if $h(G)=1$. The group $G$ is almost recognizable (resp. nonrecognizable) if $1<h(G)<\infty$ (resp. $h(G)=\infty$). In the present paper, we focus our attention on the projective general linear groups ${PGL}(2,p^n)$, where $p=2^α3^β+1$ is a prime, $α\geq 0, β\geq 0$ and $n\geq 1$, and we show that these groups cannot be almost recognizable, in other words $h({PGL}(2,p^n))\in \{1, \infty\}$. It is also shown that the projective general linear groups ${PGL}(2,7)$ and ${PGL}(2,9)$ are nonrecognizable. In this paper a computer program has also been presented in order to find out the primitive prime divisors of $a^n-1$.

math.GR↗