Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W . We use this correspondence to find all W for which L is supersolvable. In particul...

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...

We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof t...

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.

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/?).

a positive integer $n$ is called a clt number if every group of order $n$ satisfies the converse of lagrange's theorem. in this note, we find all clt and supersolvable numbers up to $1000$. we also formulate some questions about the distribution of these numbers.

There has been some interest on how the average character degree affects structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, number zeros characters group G as in table divided irreducible $. show that if < 1 then is solvable also \frac{1}{2} supersolvable. characterise abelian groups showing \frac{1}{3} only abelian.

Let [Formula: see text] be a finite group and text]. In this paper, we prove that if text], then is supersolvable. particular, some new characterizations of the well-known groups are obtained. We also show does not imply supersolvability for no constant