Abstract. We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [MT] by combining results of McNamara [Mc] and Liu [Li]. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and di...

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.

Our main result describes how to extend a matroid so that its ground set is a modular hyperplane of the larger matroid. This result yields a new way to view Dowling lattices and new results about line-closed geometries. We complement these topics by showing that line-closure gives simple geometric proofs of the (mostly known) basic results about Dowling lattices. We pursue the topic of line-clo...

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study algebras containing codimension 1 supersolvable in which such have 2, also analyze case filiform give examples to clarify some results, including listing values α β low dimensional over complex field that been classified.Communicated by Alberto Elduque