Related papers: An arithmetic algebraic regularity lemma

Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain $\Def(t)$ we only assume that $t$ is…

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the…

The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n^dim(G).

In this paper we introduce and study the concept of normality degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be normal. Explicit formulas are obtained for some particular classes of…

In this paper we describe all group gradings by an arbitrary finite group $G$ on non-simple finite-dimensional superinvolution simple associative superalgebras over an algebraically closed field $F$ of characteristic 0 or coprime to the…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this…

Let $G$ be an abelian group of bounded exponent and $A \subseteq G$. We show that if the collection of translates of $A$ has VC dimension at most $d$, then for every $\epsilon>0$ there is a subgroup $H$ of $G$ of index at most…

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

Let $\sigma$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $\sigma$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points…

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost…

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this…

We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $\sigma$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^\sigma$…

Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\hat\Phi([A])$ be the set of points having density 1 in $A$. Sets of the form $\hat\Phi([A])$ are called $\mathcal{T}$-regular. We establish several results…

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an…

Let $\Gamma$ be a graph with vertex set $V$, and let $a$ and $b$ be nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in $\Gamma$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in…

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a…