Related papers: Normal coverings of linear groups

A finite group $G$ admits a normal $2$-covering if there exist two proper subgroups $H$ and $K$ with $G=\bigcup_{g\in G}H^g\cup\bigcup_{g\in G}K^g$. For determining inductively the finite groups admitting a normal $2$-covering, it is…

Let $\mathcal{N}\mathcal{F}$ be the class of smooth non-flat curves near the origin and near infinity previously introduced by the second author and let $\gamma\in\mathcal{N}\mathcal{F}$. We show - via a unifying approach relative to the…

Let $\Gamma$ be a group acting on with finite stabilizers and finite fundamental domain on a building of type $\tilde A_2$. We prove that any non-trivial normal subgroup of $\Gamma$ is of finite index in $\Gamma$.

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $\sigma_0(G)$ the smallest size of a primary covering of $G$, and…

The covering number of a finite group $G$, denoted $\sigma(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $\sigma(G)$ for a family of primitive groups $G$ with a unique minimal…

Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation…

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $|G| = d(d+e)$. Snyder has shown that if $e > 1$, then $|G|$ is bounded by a function of $e$. This…

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of…

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer…

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a…

Let $q$ be a prime and $A$ an elementary abelian group of order at least $q^3$ acting by automorphisms on a finite $q'$-group $G$. It is proved that if $|\gamma_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of…

For a finite group $G$, denote by $\alpha(G)$ the minimum number of vertices of any graph $\Gamma$ having $\text{Aut}(\Gamma)\cong G$. In this paper, we prove that $\alpha(G)\leq |G|$, with specified exceptions. The exceptions include four…

We study the critical exponents of discrete subgroups of a higher rank semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $\Gamma< G$ is a discrete…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |\pi(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and \pi(G) is the set of…

Let $\widetilde{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of…

Let $G$ be a graph and $\Gamma$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\Gamma$, denoted by $R(G, \Gamma)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to\Gamma$…

We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation…

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $\Gamma$ be a lattice in $G$, with $\pi:G\to G/\Gamma$ the quotient map. For a semi-algebraic…