If P is a p-group for some prime p we shall write M (P ) to denote the set of all maximal subgroups of P and Md(P ) = {P1, ..., Pd} to denote any set of maximal subgroups of P such that ∩d i=1 Pi = Φ(P ) and d is as small as possible. In this paper, the structure of a finite group G under some assumptions on the c-normal or s-quasinormally embedded subgroups in Md(P ), for each prime p, and Syl...

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we p...

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...

In this paper we give convex-ear decompositions for the order complexes of several classes of posets, namely supersolvable lattices with non-zero Möbius functions and rank-selected subposets of such lattices, rank-selected geometric lattices, and rank-selected face posets of shellable complexes which do not include the top rank. These decompositions give us many new inequalities for the h-vecto...

A subalgebra $B$ of a Lie algebra $L$ is called weak c-ideal if there subideal $C$ such that $L=B+C$ and $B\cap C\leq B_{L} $ where $B_{L}$ the largest ideal contained in $B.$ This analogous to concept weakly c-normal subgroups, which has been studied by number authors. We obtain some properties c-ideals use them give characterisations solvable supersolvable algebras. also note one-dimensional ...

The class of Worpitzky-compatible subarrangements a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings characteristic Ehrhart quasi-polynomials into one formula. braid arrangement, type A, are known as graphic arrangements. We prove that arrangements characterized cocomparability graphs. This can be re...

The following paper examines properties of the weak order on the groups Sr+1 and Br+1 as motivated by the search for a combinatorial proof of known h-vector inequalities. Several general results are developed with respect to descent set domination, and an explanation of the computer programs used to generate a number of these results is presented. While this paper primarily focuses on hand flag...

Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K * M . We call the group M ∩ K/L a c-section of M . And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M . For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L2(p) or Zq, where p and q are primes, ...