Determinants of adjacency matrices of graphs
We study the set of all determinants of adjacency matrices of graphs with a given number of vertices.
math.CO↗
arXiv subjects
Alireza Abdollahi
Publications and source records attributed to Alireza Abdollahi.
Groups with maximal irredundant covers and minimal blocking sets
Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with every hyperplane of $\mathrm{PG}(n,q)$. A blocking set is called minimal if none of its proper subsets are blocking sets. In this note we prove that if $\mathrm{PG}(n_i,q)$ contains a minimal blocking set of size $k_i$ for $i\in\{1,2\}$, then $\mathrm{PG}(n_1+n_2+1,q)$ contains a minimal blocking set of size $k_1+k_2-1$. This result is proved by a result on groups with maximal irredundant covers.
math.GR↗
Non-cyclic graph associated with a group
We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $Γ$, $w(Γ)$ denotes the clique number of $Γ$, which is the maximum size (if it exists) of a complete subgraph of $Γ$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$.
math.GR↗
Characterization of SL(2,q) by its non-commuting graph
Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let $\mathrm{SL}(2,q)$ be the special linear group of degree 2 over the finite field of order $q$. In this paper we prove that if $G$ is a group such that $\mathcal{A}_G\cong \mathcal{A}_{\mathrm{SL}(2,q)}$ for some prime power $q\geq 2$, then $G\cong \mathrm{SL}(2,q)$.
math.GR↗
Non-cyclic graph of a group
We associate a graph $Γ_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left \text{is cyclic for all} y\in G\}$, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of $Γ_G$ is finite if and only if $Γ_G$ has no infinite clique. We prove that if $G$ is a finite nilpotent group and $H$ is a group with $Γ_G\congΓ_H$ and $|Cyc(G)|=|Cyc(H)|=1$, then $H$ is a finite nilpotent group. We give some examples of groups $G$ whose non-cyclic graphs are ``unique'', i.e., if $Γ_G\cong Γ_H$ for some group $H$, then $G\cong H$. In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups $G$ with the property that if $Γ_G \cong Γ_H$ for some group $H$, then $|G|=|H|$. These suggest the question whether the latter property holds for all finite non-cyclic groups.
math.GR↗
Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images
A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that every 2-generator $E$-group is abelian and that all 3-generator $E$-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator $E$-group is abelian. We conjecture that every finite 3-generator $E$-group should be abelian. Moreover we show that the minimum order of a non-abelian $pE$-group is $p^8$ for any odd prime number $p$ and this order is $2^7$ for $p=2$. Some of these results are proved for a class wider than the class of $E$-groups.
math.GR↗
Minimal blocking sets in PG(n,2) and covering groups by subgroups
In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $ =PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the latter result we show that if $G$ is a finite 2-group and $n$ is a positive integer, then $G$ admits a $\mathfrak{C}_{n+1}$-cover if and only if $n$ is even and $G\cong (C_2)^{n}$, where by a $\mathfrak{C}_m$-cover for a group $H$ we mean a set $\mathcal{C}$ of size $m$ of maximal subgroups of $H$ whose set-theoretic union is the whole $H$ and no proper subset of $\mathcal{C}$ has the latter property and the intersection of the maximal subgroups is core-free. Also for all $n<10$ we find all pairs $(m,p)$ ($m>0$ an integer and $p$ a prime number) for which there is a blocking set $B$ of size $n$ in $PG(m,p)$ such that $ =PG(m,p)$.
math.GR↗
Engel graph associated with a group
Let $G$ be a non-Engel group and let $L(G)$ be the set of all left Engel elements of $G$. Associate with $G$ a graph $\mathcal{E}_G$ as follows: Take $G\backslash L(G)$ as vertices of $\mathcal{E}_G$ and join two distinct vertices $x$ and $y$ whenever $[x,_k y]\not=1$ and $[y,_k x]\not=1$ for all positive integers $k$. We call $\mathcal{E}_G$, the Engel graph of $G$. In this paper we study the graph theoretical properties of $\mathcal{E}_G$.
math.GR↗
1-Factorizations of Cayley graphs
In this note we prove that all connected Cayley graphs of every finite group $Q \times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order.
math.CO↗
Finite groups with a certain number of elements pairwise generating a non-nilpotent subgroup
Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $ $ is in $\mathcal{X}$. Let $\mathcal{N}$ and $ \mathcal{A}$ be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If $G$ is a finite semi-simple group satisfying the condition $(\mathcal{N},n)$, then $|G|<c^{2[\log_{21}n]n^2} [\log_{21}n]!$, for some constant $c$. (2) A finite insoluble group $G$ satisfies the condition $(\mathcal{N},21)$ if and only if $\frac{G}{Z^*(G)}\cong A_5$, the alternating group of degree 5, where $Z^*(G)$ is the hypercentre of $G$. (3) A finite non-nilpotent group $G$ satisfies the condition $(\mathcal{N}, 4)$ if and only if $\frac{G}{Z^*(G)}\cong S_3$, the symmetric group of degree 3. (4) An insoluble group $G$ satisfies the condition $(\mathcal{A},21)$ if and only if $G\cong Z(G)\times A_5$, where $Z(G)$ is the centre of $G$. (5) If $d$ is the derived length of a soluble group satisfying the condition $(\mathcal{A},n)$, then $d=1$ if $n\in \{1,2\}$ and $d\leq 2n-3$ if $n\geq 2$.
math.GR↗
Left 3-Engel elements in groups
In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime $p$ and any left 3-Engel element $x$ of finite $p$-power order in a group $G$, $x^p$ is in the Baer radical of $G$. Also it is proved that $ $ is nilpotent of class 4 for every two left 3-Engel elements in a group $G$.
math.GR↗
Groupes satisfaisant une condition nilpotence
In this Note we study the groups $G$ satisfying condition $(\mathcal{N},n)$, that is, every subset of $G$ with $n+1$ elements contains a pair $\{x,y\}$ such that the subgroup $ $ is nilpotent.
math.GR↗
Certain locally nilpotent varieties of groups
Let $c\geq 0$, $d\geq 2$ be integers and $\mathcal{N}_c^{(d)}$ be the variety of groups in which every $d$-generator subgroup is nilpotent of class at most $c$. N.D. Gupta posed this question that for what values of $c$ and $d$ it is true that $\mathcal{N}_c^{(d)}$ is locally nilpotent? We prove that if $c\leq 2^d+2^{d-1}-3$ then the variety $\mathcal{N}_c^{(d)}$ is locally nilpotent and we reduce the question of Gupta about the periodic groups in $\mathcal{N}_c^{(d)}$ to the prime power finite exponent groups in this variety.
math.GR↗
A combinatorial problem in infinite groups
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, ..., X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1, ..., x_n)=1$. Clearly, $\mathcal{V}(w) \cup \mathcal{F} \subseteq \mathcal{V}(w^*)$; $\mathcal{F}$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $\mathcal{P}$ of groups for which the equality $(\mathcal{V}(w) \cup \mathcal{F})\cap \mathcal{P}= \mathcal{P} \cap \mathcal{V}(w^*)$ holds.
math.GR↗