Related papers: Optimal bounds for zero-sum cycles. I. Odd order

Let $C_{n}$ be a cycle of length $n$. As an application of Szemer\'{e}di's regularity lemma, {\L}uczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that…

A large class of orbifold quiver gauge theories admits the action of finite Heisenberg groups of the form \prod_i Heis(Z_{q_i} x Z_{q_i}). For an Abelian orbifold generated by \Gamma, the Z_{q_i} shift generator in each Heisenberg group is…

Given a finite non-cyclic group $G$, call $\sigma(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $\sigma(G)$ for $G$ a group with a unique minimal normal subgroup $N$…

In part II we constructed the lower bound, in the spirit of $\Gamma$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_\Omega…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

For a finite group $G$, let $\psi(G)$ denote the sum of element orders of $G$. This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group $G$ of order $n$, $\psi(G)$ is maximum if and only if…

Let $G$ be a finite group of order $n$, and let $C_n$ be the cyclic group of order $n$. We show that $\sum_{g \in C_n} \phi(\mathrm{o}(g))\geq \sum_{g \in G} \phi(\mathrm{o}(g))$, with equality if and only if $G$ is isomorphic to $C_n$. As…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…

Let $n$ be a positive integer and let $[n]=\{1,2,\ldots,n\}$. Let $\Gamma_n$ denote the group of permutations on $[n]$ whose restrictions to maximal proper subsets of $[n]$ are even, let $\Sigma_n$ denote the monoid of transformations on…

Let G:=SO(n,1)^\circ and \Gamma be a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(\Gamma \ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and…

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed…

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup…

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted…

We give a short new proof that for each non-elementary Kleinian group $\Gamma$, the exponent of convergence of an arbitrary non-trivial normal subgroup is bounded below by half of the exponent of convergence of $\Gamma$, and that strict…

Let $\Gamma$ be an infinite discrete subgroup of Gl$_n(\mathbb{C})$. Then either $(\mathbb{R}, <, +, \cdot, \Gamma)$ is interdefinable with $(\mathbb{R}, <, +, \cdot, \lambda^\mathbb{Z})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},…

The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. An $L( 2,1)$-labeling of graph $\Gamma$…

We introduce the vertex-arboricity of group-labelled graphs. For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose edges are labelled by elements of $\Gamma$. For an abelian group $\Gamma$ and $A\subseteq \Gamma$, the…

Let $(\Gamma,+)$ be an Abelian group of order $n^2$ and MS$_{\Gamma}(n)$ be an $n\times n$ array whose entries are all elements of $\Gamma$. Then MS$_{\Gamma}(n)$ is a $\Gamma$-magic square if all row, column, main and backward main…

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at…