Related papers: Difference bases in dihedral groups

The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $[n]=\{1,2, \ldots ,n\}$ is the graph on $[n]$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. Let $H$ be a fixed graph…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$.…

Given a permutation group $G \le \mathrm{Sym}(\Omega)$, a subset $B$ of $\Omega$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$,…

We study the relation between the Galois group $G$ of a linear difference-differential system and two classes $\mathcal{C}_1$ and $\mathcal{C}_2$ of groups that are the Galois groups of the specializations of the linear difference equation…

Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U =\{u_1,u_2,\cdots,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b \in V(G)$, there is a…

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is…

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the…

Let $(X,\S)$ be a set system on an $n$-point set $X$. The \emph{discrepancy} of $\S$ is defined as the minimum of the largest deviation from an even split, over all subsets of $S \in \S$ and two-colorings $\chi$ on $X$. We consider the…

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split Cartan subalgebra of L. Then L has a Chevalley basis with respect to H. If the characteristic of F is not 2 or 3, it is known how to find it.…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

We study the subvariety of singular sections, the discriminant, of a base point free linear system $|L|$ on a smooth toric variety $X$. On one hand we describe pairs $(X,L)$ for which the discriminant is of low dimension. Precisely, we…

When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic…

We introduce the {\it endomorphism distinguishing number} $D_e(G)$ of a graph $G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$ colors that is only preserved by the trivial endomorphism. This generalizes the notion…

Consider a random polynomial $$ G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+...+\xi_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q, ..., Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show…

A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following…

Let $G$ be a permutation group on a finite set $\Omega$. A subset $B \subseteq \Omega$ is a base for $G$ if the pointwise stabilizer of $B$ in $G$ is trivial. The base size of $G$, denoted $b(G)$, is the smallest size of a base. A well…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group…

This note considers a finite algebraic group $G$ acting on an affine variety $X$ by automorphisms. Results of Dufresne on polynomial separating algebras for linear representations of $G$ are extended to this situation. For that purpose, we…

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $\theta(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is…