Related papers: Finite groups, minimal bases and the intersection …

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In…

Let $G$ be a group. The intersection subgroup graph of $G$ (introduced by Anderson et al. \cite{anderson}) is the simple graph $\Gamma_{S}(G)$ whose vertices are those non-trivial subgroups say $H$ of $G$ with $H\cap K=\{e\}$ for some…

Given a finite permutation group $G$ with domain $\Omega$, we associate two subsets of natural numbers to $G$, namely $\mathcal{I}(G,\Omega)$ and $\mathcal{M}(G,\Omega)$, which are the sets of cardinalities of all the irredundant and…

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup…

Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such…

A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order…

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

Let $\Gamma$ be a finite group, let $\theta$ be an involution of $\Gamma$, and let $\rho$ be an irreducible complex representation of $\Gamma$. We bound $\dim \rho^{\Gamma^{\theta}}$ in terms of the smallest dimension of a faithful…

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq \Phi(G)$ and…

The biplanar crossing number of a graph $G$ is the minimum number of crossings over all possible drawings of the edges of $G$ in two disjoint planes. We present new bounds on the biplanar crossing number of complete graphs and complete…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

Let $p$ be a prime and $q = p^k$. A subset $\mathcal{F} \subset \operatorname{\Gamma L}_{2}(q)$ is intersecting if any two semilinear transformations in $\mathcal{F}$ agree on some non-zero vector in $\mathbb{F}_q^2$. We show that any…

Let $G$ be a transitive permutation group acting on $\Omega$. In this paper, we introduce and study the parameter ${\bf m}(G)$, which denotes the size of the smallest set of points $A$ such that, for every permutation $g\in G$, $A \cap A^g$…

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying…

Let A,B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group; i.e. (AB union BA) = G. The study of this event gives rise to a group invariant we call \Theta(G).…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall…

A temporal graph $G$ is a sequence $(G_t)_{t \in I}$ of graphs on the same vertex set of size $n$. The \emph{temporal exploration problem} asks for the length of the shortest sequence of vertices that starts at a given vertex, visits every…

If $g\in G$ is a non-trivial element in a residually finite group, then there exists by definition a finite group $Q$ and a homomorphism $\varphi: G \to Q$ such that $\varphi(g) \neq e$. The residual finiteness growth $\text{RF}_G$ of a…

We approximate intersection numbers $\big\langle \psi_1^{d_1}\cdots \psi_n^{d_n}\big\rangle_{g,n}$ on Deligne-Mumford's moduli space $\overline{\mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain…