The lattices and Möbius functions of stable closed subrootsystems and Hyperplane complements for classical Weyl Groups.
The local finiteness of some groups generated by a conjugacy class of order 3 elements.
The lower central series of a fiber-type arrangement.
The lower central series of the free partially commutative group.
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.