Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid factor...

Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of the characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid fa...

–Let p be a prime number, G p-solvable finite group and P Sylow p-subgroup of G. We prove that is p-supersolvable if NG(P) there subgroup H with P??H??(P) such s-semipermutable in As applications, we simplify the proofs some known results also generalize results.

For a maximal subgroup M of a finite group G , the normal index r¡(G : M) is defined to be the order of a chief factor H/K where H is minimal in the set of supplements of M in G . We obtain several results on the normal index of maximal subgroups M of composite index in G with [G : M]p = 1 which imply G to be solvable, supersolvable.

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, ...

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, ...

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...