Related papers: Midconvex sets in Abelian groups

A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size $O((n \cdot 2^n)^{1/3})$ in the group $\mathbb{Z}_2^n$, generalizing a result…

Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when…

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in…

A torsion-free abelian group B of arbitrary rank is called a B_1-group if Bext^1(B,T)=0 for every torsion abelian group T, where Bext^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a…

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the…

We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.

This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is…

An endomorphisms $\varphi$ of an abelian group $A$ is said inertial if each subgroup $H$ of $A$ has finite index in $H+\varphi (H)$. We study the ring of inertial endomorphisms of an abelian group. Here we obtain a satisfactory description…

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…

Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then…

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting.…

A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (as multisets).…

Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA \neq G$, it is called a nonbasis of order $h$. Our interest is in all possible sizes of $hA$ when $A$ is a nonbasis of order…

A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. A parallelogram, P, is a special case of a…

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two…

In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the…

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take…

A non-empty set $X$ of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by $X$ is connected and it is called convex if no two vertices of $X$ are connected by a directed path in which some…

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…

We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have…