Related papers: On balanced subgroups of the multiplicative group

A normal subgroup $E$ of a group $G$ is said to be hypercyclically embedded in $G$ if either $E=1$ or $E\neq 1$ and every chief factor of $G$ below $E$ is cyclic. In this article, we present some new characterizations of a normal subgroup…

Signed networks are graphs whose edges are labelled with either a positive or a negative sign, and can be used to capture nuances in interactions that are missed by their unsigned counterparts. The concept of balance in signed graph theory…

A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced if its graph $G(\Delta)$ is $d$-colorable. Klee and Novik obtained the balanced lower bound theorem for balanced normal $(d-1)$-pseudomanifolds $\Delta$ with $d\geq3$ by showing…

A balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We define the notion of a good pair and…

This paper addresses two open questions posed in [27] regarding the balanced domination number in graphs. We show that three new classes of graphs, those of convex polytopes A_n, D_n, and Rn'', are d-balanced. Further, we provide a…

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the…

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups,…

We describe a new type of sufficient condition for a balanced bipartite digraph to be hamiltonian. Let $D$ be a balanced bipartite digraph and $x,y$ be distinct vertices in $D$. $\{x, y\}$ dominates a vertex $z$ if $x\rightarrow z$ and…

The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $ \mathcal{E}(G)$ spanned by copies of…

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $\Gamma_H(G)$ whose vertex set $V(\Gamma_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if…

In 2008, Wang \& Wang showed that the set of gaps of a numerical semigroup generated by two coprime positive integers $a$ and $b$ is equidistributed modulo 2 precisely when $a$ and $b$ are both odd. Shor generalized this in 2022, showing…

Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$…

In this paper we investigate the following general problem. Let $G$ be a group and let $i(G)$ be a property of $G$. Is there an integer $d$ such that $G$ contains a $d$-generated subgroup $H$ with $i(H)=i(G)$? Here we consider the case…

We define the notion of a semicharacter of a group G : A function from the group to C*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is…

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard way)…

For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike…

For a group $G$, we define a graph $\Delta(G)$ by letting $G^{\#} = G \setminus \{ 1 \}$ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\#}$ if and only if the subgroup $\langle x,y\rangle$ is cyclic.…

Let $d$ be a square-free integer, $\mathbf{k}=\mathbb{Q}(\sqrt d,\,i)$ and $i=\sqrt{-1}$. Let $\mathbf{k}_1^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}$, $\mathbf{k}_2^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}_1^{(2)}$…

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism for any prime $p$. When $p - 1$ is not divisible by small integers…