Related papers: Simple groups, generation and probabilistic method…
In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in $\mathbf{SL}_2(\mathbb{R})$ from which…
We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of…
We determine for which known finite simple groups $G$ and which primes $p$ the $p$-fusion system of $G$ is simple. This means first collecting together the results that were already known (and correcting two errors made in an earlier study…
Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the…
We study `good elements' in finite $2n$-dimensional classical groups $G$: namely $t$ is a `good element' if $o(t)$ is divisible by a primitive prime divisor of $q^n-1$ for the relevant field order $q$, and $t$ fixes pointwise an $n$-space.…
This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of…
We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is…
For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…
It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…
We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups $Z_p$, and the alternating groups $Alt_{n>4}$. After a quick revision of finite fields $\mathbb{F}_q$, $q =…
For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the…
Humans spend a significant part of their lives being a part of groups. In this document we propose research directions that would make it possible to computationally form productive groups. We bring to light several issues that need to be…
Statistical and probabilistic characteristics of locally free group with growing number of generators are defined and their application to statistics of braid groups is given.
Generative Artificial Intelligence is emerging as an important technology, promising to be transformative in many areas. At the same time, generative AI techniques are based on sampling from probabilistic models, and by default, they come…
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…
In this paper we survey a new criteria for solvability of finite groups in terms of number of supersolvable (also known as polycyclic) and non-supersolvable subgroups. In particular, we present original examples of supersolvable groups such…
Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special…
Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and…
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods, group-theoretic and coming from algebraic and arithmetic…