Maximal and Frattini L-subgroups of an L-group

Abstract

In this paper, the concept of a maximal L-subgroup of an L-group has been defined in the spirit of classical group theory. Then, a level subset characterization has been established for the same. Then, this notion of maximal L-subgroups has been used to define Frattini L-subgroup. Further, the concept of non-generators of an L-group has been developed and its relation with the Frattini L-subgroup of an L-group has been established like their classical counterparts. Moreover, several properties pertaining to the concepts of maximal L-subgroups and Frattini L-subgroup have also been investigated. These two notions have been illustrated through several examples.

An Application of Maximality to Nilpotent and Finitely Generated L-Subgroups of an L-Group

“…The variable µ is an L-subgroup of G if and only if µ is an L-subgroup of 1 G . Definition 2.8 ( [12]). Let η ∈ L(µ) such that η is nonconstant and η = µ.…”

“…Conversely, if η (n) = η for some nonnegative integer n, then the series of L-subgroups 1) . Now, we recall the definition of a maximal L-subgroup of an L-group from [13]: Definition 3.14. Let µ ∈ L(G).…”

“…In this section, we explore some properties of the maximal and Frattini L-subgroups of the finitely generated L-subgroups. The Frattini L-subgroup of an L-group was investigated by Jahan and Manas [13]. Moreover, the authors defined the nongenerators of an L-group and established a relationship between the non-generators and the Frattini L-subgroup of µ.…”

“…However, these studies have been inadequate for L-groups. Recently, various concepts of classical group theory have been extended to L-settings, particularly considering their compatibility [8][9][10][11][12]. Moreover, in [13], the authors investigated the maximal and Frattini L-subgroups of an L-group.…”

“…Recently, various concepts of classical group theory have been extended to L-settings, particularly considering their compatibility [8][9][10][11][12]. Moreover, in [13], the authors investigated the maximal and Frattini L-subgroups of an L-group. The notion of non-generators of an L-group was also introduced, and a relationship between the Frattini L-subgroup and the set of non-generators of an L-group was established.…”