The Magic World of Geometry - II. Geometry and Algebra of Braids.
The Magic World of Geometry - III. The Dirac String Problem.
The Malnormality of Certain Subgroups of Small Cancellation Groups.
The mapping class group of a genus two surface is linear.
The Minimal 5-Representation of Lyon's Sporadic Group.
The minimizing of the Nielsen root classes
Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen...
The mixed product decompositions of partially ordered groups
The th root of a braid is unique up to conjugacy.
The nil Hecke ring and Deodhar's conjecture on Bruhat intervals.
The nilpotency of some groups with all subgroups subnormal.
Let G be a group with all subgroups subnormal. A normal subgroup N of G is said to be G-minimax if it has a finite G-invariant series whose factors are abelian and satisfy either max-G or min- G. It is proved that if the normal closure of every element of G is G-minimax then G is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.
The number of sides of a parallelogram.
The Number of Solutions of x3 = 1 in a 3-Group.
The orbits of the Hurwitz action of the braid groups on the standard generators
The Hurwitz action of the n-braid group Bₙ on the n-fold direct product of the m-braid group is studied. We show that the orbit of any n- tuple of the n standard generators of consists of the (n-1)th powers of n+1 elements.
The order dimension of Bruhat order on infinite Coxeter groups.
The order of a finite Coxeter group.
The Ore conjecture
The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character-theoretic methods with other ingredients, and use them to establish the conjecture.
The -central extension of the Mapping Class Group of orientable surfaces
Topological Quantum Field Theories are closely related to representations of Mapping Class Groups of surfaces. Considering the case of the TQFTs derived from the Kauffman bracket, we describe the central extension coming from this representation, which is just a projective extension.
The p-adic Topology on a Free Group: A Counterexample.
The periodic groups saturated by finitely many finite simple groups.
The Poisson formula for groups with hyperbolic properties.