Buildings and shadows.
-groups.
Characterization of finite groups by their commuting graph.
Characterization of -solvable groups.
Characterization of by its non-commuting graph.
Characterization of the alternating groups by their order and one conjugacy class length
Let be a finite group, and let be the set of conjugacy class sizes of . By Thompson’s conjecture, if is a finite non-abelian simple group, is a finite group with a trivial center, and , then and are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In...
Commutative subloop-free loops
We describe, in a constructive way, a family of commutative loops of odd order, , which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group .
Construction of 2-local finite groups of a type studied by Solomon and Benson.
Correction to the paper “Generators of an orthogonal group over a finite field”
Coset enumeration of groups generated by symmetric sets of involutions.
Crossed product of cyclic groups
All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.
(p ≥ 5) can be characterized by its order components
Let G be a finite group, and (p ≥ 3). It is proved that G ≅ M if G and M have the same order components.
Détermination des caractères des groupes finis simples : travaux de Lusztig
Disconnected groups of Lie type as Galois groups.
Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
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 geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
Exceptional groups of Lie type as Galois groups.
Expansion in finite simple groups of Lie type
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].
Factorizations of Finite Groups.
Finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index.
Finite simple groups of Lie type as expanders
We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.