Related papers: Zero-sum problems with congruence conditions

We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various…

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each…

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge…

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when…

Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a $G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$ with the property that $f(v)=0$ for every positive degree homogeneous invariant $f$ in…

Let $r$, $m$ and $k\geq 2$ be positive integers such that $r\mid k$ and let $v \in \left[ 0,\lfloor \frac{k-1}{2r} \rfloor \right]$ be any integer. For any integer $\ell \in [1, k]$ and $\epsilon \in \{0,1\}$, we let…

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism $\varphi$, an element $x\in G$ and a subset $K\subseteq G$, we say that the relative $\varphi$-order of $g$ in $K$,…

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…

Let $m$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}_m)$ is the least integer $N$ (if it exists) such that for every edge-coloring $\chi \, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_m$ one can find…

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind(S)$ of $S$ is defined to be the…

Let $G$ be a graph with degree sequence $d_1\geq \ldots \geq d_n$. Slater proposed $s\ell(G)=\min\{ s: (d_1+1)+\cdots+(d_s+1)\geq n\}$ as a lower bound on the domination number $\gamma(G)$ of $G$. We show that deciding the equality of…

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$\rho_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and…

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of…

Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem…

For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from…

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. Let $n_D$ be the smallest non-negative integer for which the following statement holds: if $C$ is a $p$-divisible group over $k$ of…

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 $<x,y>$ is…