Related papers: A note on the probability of a groupoid having def…
Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, the general formula for computing the $P(\mathbb{F}_qG)$ are…
The divergence of a group is a quasi-isometry invariant defined in terms of pairs of points and lengths of paths avoiding a suitable ball around the identity. In this paper we study "random divergence'', meaning the divergence at two points…
Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…
If $H$ is a subgroup of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $G$ is denoted by $Pr(H,G)$. Let $\langle g\rangle$ stand for the monothetic subgroup generated by an element $g\in…
This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete…
We show that for any finite group $G$ and for any $d$ there exists a word $w\in F_{d}$ such that a $d$-tuple in $G$ satisfies $w$ if and only if it generates a solvable subgroup. In particular, if $G$ itself is not solvable, then it cannot…
Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most…
The equational probabilistic spectrum of a finite algebra is the set of probabilities with which equations are satisfied in the algebra. We study algebras with minimal spectrum, that is, spectra consisting only of the values $1$ and…
A countable discrete group is called Choquet-Deny if for any non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Building on the previous work of Jaworski, a complete…
The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y :…
A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\H os implies that…
Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq…
The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set…
In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the…
A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length…
This paper examines order three elements of finite groups which normalize no nontrivial 2-subgroup. The motivation for finding such elements arises out of a problem in modular representation theory. The question of when these elements…
The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given element of that ring. We obtain several…
We give a necessary condition of generic 3 -rigidity of graphs relying on partitioning the edges into 3 subsets; such that each subset-pair gives a generically 2-rigid graph, either by themselves or after an appropriate edge-deletion.…
Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary…
We consider the group $\mathfrak{X}(G)$ obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. Deceptively complicated finitely presented groups arise from…