**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.

**Keywords:** L-Subgroup, Generated L-subgroups, Commutator L-subgroup, Characteristic L-subgroup, Nilpotent L-subgroup, Solvable L-subgroup, supersolvable L-subgroup, Subnormal L-subgroup,
