Related papers: Imprimitive Permutations in Primitive Groups

We survey a research program on the strong convergence of unitary and permutation representations of discrete groups. We also take the opportunity to flesh out details that have not appeared elsewhere.

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of…

This paper studies simplicity, primitivity and semiprimitivity of algebras associated to \'etale groupoids. Applications to inverse semigroup algebras are presented. The results also recover the semiprimitivity of Leavitt path algebras and…

Primitive representations of finite groups as well as primitive finite groups were classified in the O'Nan-Scott Theorem. In this paper we classify faithful finite primitive semigroup representations. To each finite primitive…

We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…

The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the…

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…

Our aim is to describe the theory of Cartesian decompositions preserved by some member of a large family of finite transitive permutation groups called innatelytransitive groups.

We present a representation for permutation groups as the automorphism group of an ordered set $U$ such that the automorphism group's action on a subset $T\subseteq U$ is the permutation group itself. For many imprimitive permutation…

Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…

This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on…

We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study…

Let $G$ be a transitive permutation group acting on a finite set $\Omega$ with $|\Omega|\geqslant 2$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$, and by a theorem of Jordan from 1872, $G$ always…

A base B for a finite permutation group G acting on a set X is a subset of X with the property that only the identity of G can fix every point of B. We prove that a primitive diagonal group G has a base of size 2 unless the top group of G…

In the world, in which acceptance and the identification with social communities are highly desired, the ability to predict evolution of groups over time appears to be a vital but very complex research problem. Therefore, we propose a new,…

We prove lower bounds of the form $\gg N/(\log N)^{3/2}$ for the number of primes up to $N$ primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an…

We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of…

This book is an introduction to a fast developing branch of mathematics - the theory of representations of groups. It presents classical results of this theory concerning finite groups.

We consider the problem of determining whether the union of two infinite matroids is a matroid. We introduce a superclass of the finitary matroids, the nearly finitary matroids, and prove that the union of two nearly finitary matroids is a…

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative…