Related papers: The Spread of Almost Simple Classical Groups

This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…

A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in…

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of…

In this short note we answer to a question of group theory from arXiv:0910.5080. In that paper the author describes the set of realizable Steinitz classes for so-called $A'$-groups of odd order, obtained iterating some direct and semidirect…

We recall Schur's work on universal central extensions and develop the analogous theory for categorical extensions of groups. We prove that the String 2-groups are universal in this sense and study in detail their restrictions to the finite…

In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these…

In this article, we study $2$-designs with $\lambda=2$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type, and we prove that such a $2$-design does not exist. In conclusion,…

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture was first proposed by German mathematician Christian Goldbach in 1742 and, despite being obviously true,…

Let $G$ be a finite almost simple group and let $H$ be a Sylow $p$-subgroup of $G$. As a special case of a theorem of Zenkov, there exist $x,y \in G$ such that $H \cap H^x \cap H^y = 1$. In fact, if $G$ is simple, then a theorem of Mazurov…

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of…

The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute and which do not. We show that the graph product of quasi-lattice ordered groups is…

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary…

We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.

We study simplicity of $C^*$-algebras arising from self-similar groups of $\mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A. Sims in…

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum…

We follow in this paper a recent line of work, consisting in characterizing the periodically rigid finitely generated groups, i.e., the groups for which every subshift of finite type which is weakly aperiodic is also strongly aperiodic. In…

We show that for every positive integer $n$ there exists a simple group that is of type $\mathrm{F}_{n-1}$ but not of type $\mathrm{F}_n$. For $n\ge 3$ these groups are the first known examples of this kind. They also provide infinitely…

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of…

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions.…

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G…