**1. Introduction**

After Rosenfeld [1] introduced his notion of fuzzy subgroups of an ordinary group, there was a great activity in the investigations of fuzzy algebraic structures. Various aspects of fuzzy algebra were explored. Specially in the field of fuzzy group theory, several researchers contributed towards development [2–20]. However, the researchers were not coherent and their studies suffered from one kind of incompatibility or the other. In the year 1981, Liu [21] introduced his notion of normality of a fuzzy subgroup in an ordinary group. Soon after that Wu [22] came up with an idea and just gave a hint of pursuing studies of fuzzy subgroups of a fuzzy group by providing the definition of a fuzzy normal subgroup in a fuzzy group. Afterwards, this idea was further taken up only by Martinez [23, 24]. Following this approach, the theory of *L*-subgroups is developed in a systematic and consistent manner in our papers [10, 25–34].

In fact, *L*-group theory came into existence as an answer to the problems posed by fuzzy group theory and the theories of other fuzzy algebraic structures. The development of fuzzy group theory was hampered mainly because of two reasons. Firstly, it was due to some inherent problems that the analogs of some of the concepts and results of classical group theory were not even formulated in this theory (for details,

see Section 7, Discussion and Analysis). Moreover, various types of series such as derived series, descending central series, normal closure series etc. could not be formulated. Secondly, since most of the concepts studied in fuzzy algebraic structures were generically defined, Tom Head [35] proposed his well known metatheorem (which is based on the concept of Rep function) to extend the results of classical algebra to fuzzy setting. Therefore, the results of fuzzy group theory became simple instances of an application of metatheorem. Throughout the development of *L*-group theory, the above two drawbacks of fuzzy group theory have been very well taken care of (for details, see Section 7, Discussion and Analysis) and we are in a position to put forward a theory parallel to classical group theory. Thus a consistent theory came into existence. The subject matter discussed in this chapter, in particular; the join problem of *L*-subgroups provides sufficient testimony to its success [10].

*As an application and motivation, here we mention that if we replace the lattice L, in our work by the closed unit interval* ½ � 0, 1 *, then we retrieve the corresponding version for fuzzy group theory. Also, as an application of this theory we mention that if we replace the lattice L by the two elements set* f g 0, 1 *, then the results of classical group theory follow as simple corollaries of the corresponding results of L-group theory. Moreover, this development of L-group theory is beyond the purview of metatheorem, contrary to the development of fuzzy group theory*.

Section 2 provides a list of all the basic definitions and results regarding *L*-subsets and *L*-algebraic substructures which are required for the development of the subsequent sections. For the sake of completeness, few definitions from lattice theory have also been incorporated. In papers [10, 25–34], various concepts of *L*-group theory have been explored.

Section 3 introduces the concept of a normalizer of an *L*-subgroup of an *L*-group. In Subsection 3.1, this notion of normalizer is very carefully formulated by using the concept of a coset by an *L*-point of an *L*-group. The normalizer of an *L*-subgroup in an *L*group, formulated in this work, is an *L*-subgroup of the given *L*-group. This concept of classical group theory was left untouched during the evolution of fuzzy group theory. Although, Mukherjee and Bhattachrya [36, 37] tried to introduce a notion of normalizer of a fuzzy subgroup of an ordinary group, but it turned out to be a crisp subgroup of the given ordinary group. This idea was followed by several researchers [16, 38] in the past, but they were unable to obtain the results, presented in this work, due the fact that they carried out their researches within the framework where the parent structure was an ordinary group. We have shown in our work that for a normal *L*-subgroup of an *L*group, its left and right *L*-cosets are identical. Also, it has been proved that each *L*-subset of an *L*-group commutes with every normal *L*-subgroup of the given *L*-group. In the end of this subsection, we state certain properties and the nature of this normalizer under the action of a group homomorphism. In Subsection 3.2, a universal construction of a generated *L*-subgroup by an *L*-set has been provided and its relationship with level subsets is investigated. This construction along with the construction of commutator *L*-subsets, studied in Section 4, allows us to define a commutator *L*-subgroup. In Subsection 3.3, again by replacing the parent structure of an ordinary group by an *L*-subgroup, we formulate the concept of a characteristic *L*-subgroup of an *L*-group. After obtaining the level subset characterizations of this concept, we establish some group theoretic analogs. Then, we construct various types of lattices and sublattices of characteristic *L*-subgroups. Finally, in this section we are able to establish that a characteristic *L*-subgroup of a normal *L*-subgroup is normal. Also, the well known property of transitivity of characteristic subgroup of classical group theory is extended to the *L*-setting. However, the same could not be even formulated in the works of earlier researchers [14, 39–41].

### *Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

Section 4 starts with the notion of a commutator *L*-subset and commutator *L*-subgroup of an *L*-group. It is worthwhile to mention here that, earlier Gupta and Sarma [8] extended the notion of commutator subgroups in fuzzy setting which was utilized to formulate the concepts of descending central chain and derived chain in their further studies [9, 10]. However, the above mentioned studies have been carried out within the framework where the underlying group is an ordinary group. Therefore, they lost certain compatibility with other fuzzy algebraic notions. Here, we obtain some group theoretic analogs of commutator subgroups, we state a property of infimums of the set product of two *L*-subsets which is used in the further development of the subject matter. The whole development is justified by the level subset and strong level subset characterizations of commutator *L*-subgroups. Further in this section, we introduce the concept of a descending central chain of an *L*-subgroup by making the use of the notion of commutators. Then this, in turn, is used to define the notion of nilpotent *L*-subgroups of an *L*group. Here the concept of the trivial *L*-subgroup of an *L*-subgroup comes into play. The members of the descending central chain are normal *L*-subgroups in their preceding ones in the sense of Wu [22]. Then, we present some peculiarities of *L*-setting which will be discussed in the end of this chapter. The level subset and strong level subset characterizations of these notions justify these extensions. The concept of the central chain of an *L*subgroup is also introduced with the help of the trivial *L*-subgroup of the given *L*-group which is followed by analogs of some well known results of classical group theory. In the end of this section, we establish a necessary and sufficient condition for the set product of two trivial *L*-subgroups to be a trivial *L*-subgroup (see Theorem 1.81). Finally, this result has been used very effectively to establish a sufficient condition for the set product of two nilpotent *L*-subgroups to be nilpotent [31]. It has also been shown that the notion of a normalizer of an *L*-subgroup, which has been introduced in Section 3, is compatible with the notion of nilpotent *L*-subgroups. That is, nilpotent *L*-subgroup satisfies normalizer condition [34]. On the other hand, Kim [13] also defined an ascending series of crisp subgroups of an ordinary group to introduce his concept of nilpotent fuzzy subgroups. However, this could not lead to any substantial progress.

Section 5 deals with solvability and supersolvability of *L*-subgroups. Some more researchers [38, 42, 43] also discussed the notion solvability in the fuzzy setting. In the studies carried out by these authors the concept of normality introduced by Liu [21] is used. Consequently, the parent structure in their studies is an ordinary group, not a fuzzy group. For this purpose we introduce the concepts of derived series and solvable series with the help of the notion of commutator *L*-subgroups. This is possible because we use the normality in the sense of Wu [22] rather than Liu [21]. We discuss some results pertaining to the members of the derived central chain which are peculiarities of *L*-setting (Theorem 1.84, Theorem 1.85). All these concepts are justified by their level subset and strong level subset characterizations. Moreover, solvability is also characterized in terms of solvable series and some group theoretic analogs are obtained. Finally, the concept of central series is used to establish the inter connection of nilpotency and solvability of *L*subgroups. We also discuss the behavior of homomorphic and inverse homomorphic images of solvable *L*-subgroups. Next, we define normal and subinvariant *L*-subgroups of an *L*-subgroup with Abelian factors. In case when the lattice *L* is a dense chain, we characterize solvability of an *L*-subgroup with the help of a normal series with Abelian factors or subinvariant series with Abelian factors. This characterization motivated 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, commutator *L*-subgroup of a supersolvable *L*-subgroup is shown to be nilpotent. In the last, we extend Zassenhaus Theorem to *L*-setting and utilize it to establish a version of Schreier Refinement Theorem.

Section 6 evolves the concept of conjugacy in *L*-group theory. Firstly, the conjugate of an *L*-subgroup by an *L*-subgroup has been defined. The normal closure of an *L*-subgroup is defined as the *L*-subgroup generated by its conjugate by the whole parent *L*-group. Then, by using successive normal closures, we transfinitely define a series called the normal closure series of the given *L*-subgroup. Earlier an attempt has been made in [5] to define normal closure of a fuzzy group in an ordinary group. In fact, this is the fuzzy subgroup generated by the union of all the conjugates of the given fuzzy subgroup by crisp points. The concept of this conjugacy comes from [36]. But this idea was not found suitable enough to formulate the successive normal closures and hence it was not found suitable to be applied in the development of subnormal fuzzy subgroups introduced by the same author in [44]. Here, it has been shown that the normal closure series, defined in this work, is the fastest descending normal series containing given *L*-subgroup. This sets the ground for the development of subnormality in *L*-group theory [10, 33]. During the course of the development of subnormality, it has also been proved that every *L*-subgroup of a nilpotent *L*-subgroup is a subnormal *L*-subgroup. Finally in order to show the reach of *L*-group theory, we tackle the well known join problem of subnormal subgroups in *L*-setting and solve it to the same degree of success as that of classical group theory.
