in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability‎ ‎of the whole group‎.

For a hyperplane arrangement in real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by Varchenko-Gel'fand ring, which ring functions from chambers to integers with pointwise addition and multiplication. Varchenko Gel'fand gave simple presentation for this along filtration associated graded whose Hilbert series polynomial. We g...

Let B be a subset of the set of all isomorphism classes of finite groups. We consider the number Fg(x) of positive integers n < x such that all groups of order n lie in B. When ir consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for Fn(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable gr...

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...

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Mo bius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a ...

A result of Roquette 3] states that if D is an absolutely irreducible representation of a p-group G over the eld of complex numbers, then D can be realized in K((g) j g 2 G), where is the character of D and K = Q or K = Q(i) according to whether p 6 = 2 or p = 2. Based on Baum and Clausen's 1] algorithm for computing the irreducible representations of supersolvable groups, we give an elementary...

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multi...

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 ...