For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements.

Recently, it has been shown that the ordinary irreducible representations of a supersolvable group G of order n given by a power-commutator presentation can be constructed in time 0(n2 log«). We present an improved algorithm with running time 0(n log/?).

Let σ be a permutation of the set {1,2, — -, n} and let Π(N) denote the lattice of partitions of {1,2, •••,%}. There is an obvious induced action of σ on Π(N); let Π(N)σ — L denote the lattice of partitions fixed by σ. The structure of L is analyzed with particular attention paid to ^f, the meet sublattice of L consisting of 1 together with all elements of L which are meets of coatoms of L. It ...

A subalgebra B of a Leibniz algebra L is called weak c-ideal if there subideal C such that L=B+C and B∩C⊆BL where BL the largest ideal contained in B. This analogous to concept weakly c-normal subgroup, which has been studied by number authors. We obtain some properties c-ideals use them give characterizations solvable supersolvable algebras generalizing previous results for Lie algebras. note ...

Stanley [18] introduced the notion of a supersolvable lattice, L, in part to combinatorially explain the factorization of its characteristic polynomial over the integers when L is also semimodular. He did this by showing that the roots of the polynomial count certain sets of atoms of the lattice. In the present work we define an object called an atom decision tree. The class of semimodular latt...

Let L be a supersolvable lattice with nonzero Möbius function. We show that the order complex of any rank-selected subposet of L admits a convex-ear decomposition. This proves many new inequalities for the h-vectors of such complexes, and shows that their g-vectors are M-

Abstract Let p be a prime and let G finite group such that the smallest divides | is . We find sharp bounds, depending on , for commuting probability average character degree to guarantee nilpotent or supersolvable.