Development of L-Group Theory

Abstract

In this work, we present a systematic and successful development of L-group theory. A universal construction of a generated L-subgroup has been provided by using level subsets of given L-subsets. This construction allows us to define and study commutator L-subgroups, normalizer of an L-subgroup, nilpotent L-subgroups, solvable L-subgroups, normal closure of an L-subgroup. All these concepts and their inter-relationships have been presented. Here we mention that in this work we also exhibit a characterization of solvable L-subgroup with the help of a series of L-subgroups such that at each level, the factor groups of level subgroups of their consecutive members are Abelian. This allows us to introduce the notion of a supersolvable L-subgroup by using the factors of level subgroups at each level of a subinvariant series of an L-subgroup. Also, by using successive normal closures, we transfinitely define a series called the normal closure series of the L-subgroup. It has been shown that it is the fastest descending normal series containing given L-subgroup. This sets the ground for the development of subnormality in L-group theory. In the last, we study the notion of subnormal L-subgroups.