Related papers: Algorithms determining finite simple images of fin…

We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie…

Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We…

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of…

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group…

A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They include finite groups, finite posets and free categories of finite quivers as special cases. In this paper we…

A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twisted) rank $>1$, other than any $^2F_4(q)$, over a field of known size, a Las Vegas algorithm is given to produce a constructive isomorphism.…

We propose an algorithm which for any recursive group $G$, given by its effectively enumerable generators and recursively enumerable relations, outputs an explicit embedding of $G$ into a finitely presented group directly written by its…

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it…

We introduce a new constructive recognition algorithm for finite special linear groups in their natural representation. Given a group $G$ generated by a set of $d\times d$ matrices over a finite field $\mathbb{F}_q$, known to be isomorphic…

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the…

We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of…

A group $G$ given by a presentation $G = < \mathcal A \| \mathcal R >$ is called weakly finitely presented if every finitely generated subgroup of $G$, generated by (images of) some words in $\mathcal A^{\pm 1}$, is naturally isomorphic to…

We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under…

Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as $\F_q$, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its…

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this…

We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present…