EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 2086-2115 ISSN 1307-5543 – ejpam.com Published by New York Business Global Subnormality and Theory of L-subgroups Naseem Ajmal1, Ifffat Jahan2,∗, Bijan Davvaz3 1 Department of Mathematics, Zakir Husain Delhi College, University of Delhi, Delhi, India 2 Department of Mathematics, Ramjas College, University of Delhi, Delhi, India 3 Department of Mathematics, Yazd University, Yazd, Iran Abstract. The main focus in this work is to establish that L-group theory, which uses the lan- guage of functions instead of formal set theoretic language, is capable of capturing most of the refined ideas and concepts of classical group theory. We demonstrate this by extending the notion of subnormality to the L-setting and investigating its properties. We develop a mechanism to tackle the join problem of subnormal L-subgroups. The conjugate L-subgroup as is defined in our previous paper [4] has been used to formulate the concept of normal closure and normal closure series of an L-subgroup which, in turn, is used to define subnormal L-subgroups. Further, the concept of subnormal series has been introduced in L-setting and utilized to establish the subnor- mality of L-subgroups. Also, several results pertaining to the notion of subnormality have been established. Lastly, the level subset characterization of a subnormal L-subgroup is provided after developing a necessary mechanism. Finally, we establish that every subgroup of a nilpotent L- group is subnormal. In fact, it has been exhibited through this work that L-group theory presents a modernized approach to study classical group theory. 2020 Mathematics Subject Classifications: 11E57, 20N25, 20A10 Key Words and Phrases: L-subgroup, Normal L-subgroup, Nilpotent L-subgroup, Normal closure, Subnormal subgroup, Subnormal series 1. Introduction The notion of subnormality in classical group theory is an important generalization of the notion of normality. This notion is introduced to repair the deficiency of the notion of normality that it fails to be a transitive relation. In fact, the relation of subnormality can be defined as the transitive closure of the relation of normality. The class of subnor- mal subgroups not only properly contains the class of normal subgroups but in the class of finitely generated groups this class coincides with the important subclass of nilpotent subgroups. Besides, this notion has many more pleasing properties which have been the focus of attention of algebraist for a long period of time. The work of Wielandt in the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4548 Email addresses: ij.umar@yahoo.com (I. Jahan), nasajmal@yahoo.com (N. Ajmal), davvaz@yazd.ac.ir (B. Davvaz) https://www.ejpam.com 2086 © 2022 EJPAM All rights reserved. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2087 year 1939 on the join of two subnormal subgroups in a finite group provided an impetus for the development of group theory. The question of finding necessary and sufficient con- ditions for the join of subnormal subgroups to be subnormal is one of the most important problems in the area of group theory even today. Here in the Part I of this work, we introduce and study the notion of subnormality in L-group theory with a special attention of approaching the join problem of subnormal L-subgroups. This study is carried out in detail in Part II of this work and we establish necessary and sufficient conditions for the subnormality of the join of two subnormal L-subgroups of an arbitrary group. In the end of this paper, we present the main result of Part II without proof. In this paper, we present a systematic and successful development of L-group theory which has been made in papers [2–7, 10]. The important concepts of nilpoent L-subgroups, solvable L-subgroups have been introduced and studied and their inter-relationship is es- tablished in [3, 6]. Moreover, the concept of normalizer of an L-subgroup in an L-group is formulated in [2]. It has been proved in [10] that a nilpotent L-subgroup of an L-group satisfies the normalizer condition. As an application and motivation, here we mention that if we replace the lattice L, in our work by the closed unit interval [0, 1], then we retrieve the corresponding version of fuzzy group theory. Moreover, as an application of this theory we also mention that if we replace the lattice L by the two elements set {0, 1}, then the results of classical group the- ory 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. The development of fuzzy group theory could not be sustained due to some inherent problems and the emergence of metatheorem formulated by Head [9]. The hurdles faced by the researchers in this development are removed in L-group theory by considering the parent structure as an L-group rather than an ordinary group. Many higher concepts of classical group theory such as supersolvability, nilpotency and subnormality could not be studied in the framework of fuzzy group theory. The above mentioned modification allowed the formation of normalizer of an L-subgroup of an L-group similar to that of classical group theory.The question of formulation of the notion of subnormality in L- group theory leads to the question of formation of normal closure of an L-subgroup in an L-group. The concept of normal closure of an L-subgroup in an L-group should be defined in such a manner that the normal closure series arising by this normal closure should be compatible with a subnormal series arising by the given L-subgroup likewise its classical counterparts. For this purpose, we define the conjugate of an L-subset by an L-subset of an L-group instead of a crisp point. Then, the L-subgroup generated by the conjugate of an L-subgroup by the given parent L-group is called the normal closure of the given L-subgroup. This notion satisfies most of the desirable properties related to this concept. For example, the compatibility of this notion with that of commutator L-subgroup of an L-group exists like its counterpart in classical group theory. Then, this notion is utilized N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2088 in the formation of normal closure series of an L-subgroup of an L-group which leads to the definition of subnormal L-subgroup. The notion of subnormal series for an L- subgroup is introduced which exihibits the desired relationship with the normal closure series. In the end, we establish the inter-connection of subnormality of an L-subgroup with the subnormality in the usual group theoretic sense of the level subsets of the given L-subgroup by providing a result which is called the level subset characterization. Finally, by an application of this result we establish that every subgroup of a nilpotent L-subgroup is subnormal. Although so much has been said and done in the area of L-group theory, but still researchers from all over the world are involved in investigating various types of struc- tures in the fuzzy environments. A significant attempt made in this direction is due to A. Razzaque and A. Razaq [12] wherein they have defined and studied a class of fuzzy struc- ture which properly contains the classes of intuitionistic fuzzy subgroups and Pythagorean fuzzy subgroups as its subclasses. 2. Preliminaries Throughout this paper L = ⟨L,≤,∨,∧⟩ denotes a complete and completely distributive lattice where ‘≤’ denotes the partial ordering of L, the join (sup) and the meet (inf) of the elements of L are denoted by ‘∨’ and ‘∧’ respectively. Also, we write 1 and 0 for maximal and minimal elements of L respectively. The definition of a completely distributive lattice is well known in the literature and can be found in any standard text on the subject. If {Ji : i ∈ I} is any family of subsets of a complete lattice L, let F denote the set of choice functions for Ji, i.e. functions f : I → ∏ i∈I Ji such that f(i) ∈ Ji for each i ∈ I. Then, we say that L is a completely distributive lattice, if∧{∨ i∈I Ji } = ∨ f∈F {∧ i∈I f(i) } . The above law is known as the complete distributive law. Moreover, a lattice L is said to be infinitely meet distributive if for every subset {bβ : β ∈ B} of L, we have: a ∧  ∨ β∈B bβ  = ∨ β∈B {a ∧ bβ}, provided L is join complete. The above law is known as the infinitely meet distributive law. The definition of infinitely join distributive lattice is dual to the above definition. A complete lattice which satisfies infinitely meet distributive law is known as a complete Heyting algebra or a frame.Therefore, a completely distributive lattice is always a complete Heyting algebra. Further, we introduce some basic definitions and results which are used in the sequel. For details we refer to [1, 3, 8, 14? , 15] . An L-subset of X is a function from X into L. The set of L-subsets of X is called the L-power set of X and is denoted by LX . For µ ∈ LX , the tip of µ is defined as ∨ x∈X {µ(x)}. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2089 If µ, ν ∈ LX , then we say that µ is contained in ν if µ(x) ≤ ν(x) for every x ∈ X and is denoted by µ ⊆ ν. For a family {µi : i ∈ I} of L-subsets of X, where I is a non-empty index set, the union ⋃ i∈I µi and the intersection ⋂ i∈I µi of {µi : i ∈ I} are respectively defined by: (⋃ i∈I µi ) (x) = ∨ i∈I {µi(x)} and (⋂ i∈I µi ) (x) = ∧ i∈I {µi(x)}, for each x ∈ X. Let f be a mapping from a set X to a set Y . If µ ∈ LX and ν ∈ LY , then the image f(µ) of µ under f and the preimage f−1(ν) of ν under f are L-subsets of Y and X respectively, defined by f(µ)(y) = ∨ x∈f−1(y) {µ(x)} and f−1(ν)(x) = ν(f(x)). Here we point out that in the above definition if f−1(y) = ϕ, then f(µ)(y), being the least upper bound of the empty set, is zero. The set product µ ◦ ν of µ, ν ∈ LS where S is a groupoid, is an L-subset of S defined by µ ◦ ν(x) = ∨ x=yz {µ(y) ∧ ν(z)}. If µ ∈ LX and a ∈ L, then the notions of level subset µa and strong level subset µ> a of µ are, respectively, defined by: µa = {x ∈ X : µ(x) ≥ a} and µ> a = {x ∈ X : µ(x) > a}. Clearly, if a ≤ b for a, b ∈ L, then µb ⊆ µa and µ> b ⊆ µ> a . Proposition 1. [15] Let µ, ν ∈ LX and µ ⊆ ν. Then, (i) if µ ⊆ ν, then µa ⊆ νa for each a ∈ L, (ii) if µa ⊆ νa for each a ∈ Imµ, then µ ⊆ ν, (iii) if µ ⊆ ν, then µ> a ⊆ ν>a for each a ∈ L provided L is a chain, (iV) if µ> a ⊆ ν>a for each a ∈ Imν, then µ ⊆ ν provided L is a chain. Throughout this paper G denotes an ordinary group with the identity element ‘e’ and I denotes a non-empty index set. For A ⊆ X, 1A denotes the characteristic function of A in X. Definition 1. [15] Let µ ∈ LG. Then, µ is called an L-subgroup of G if for each x, y ∈ G (i) µ(xy) ≥ µ(x) ∧ µ(y), (ii) µ(x−1) = µ(x). N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2090 The set of L-subgroups of G is denoted by L(G). Clearly, the tip of an L-subgroup is attained at the identity element ‘e’ of G. Theorem 1. Let µ ∈ LG with tip a0. Then, (i) µ ∈ L(G) if and only if µa is a subgroup of G for each a ≤ a0, (ii) µ ∈ L(G) if and only if µ> a is a subgroup of G for each a < a0. It is well known in the literature that the intersection of an arbitrary family of L- subgroups of a group is an L-subgroup. Definition 2. Let µ ∈ LG. Then, the L-subgroup of G generated by µ is defined as the smallest L-subgroup of G which contains µ. It is denoted by ⟨µ⟩. That is ⟨µ⟩ = ⋂ i∈I {µi ∈ L(G) : µ ⊆ µi}. Definition 3. Let µ ∈ L(G). Then, µ is called an L-normal subgroup of G if for all x, y ∈ G, µ(xy) = µ(yx). Definition 4. If µ, η ∈ LX and η ⊆ µ, then we say that η is an L-subset of µ. The set of L-subsets of µ is denoted by Lµ. Definition 5. If µ, η ∈ L(G) and η ⊆ µ, then we say that η is an L-subgroup of µ.The set of L-subgroups of µ is denoted by L(µ). Hence onwards µ denotes an L-subgroup of G and we shall call the parent L-subgroup µ simply an L-group. Theorem 2. Let η ∈ Lµ with tip a0. Then, (i) η ∈ L(µ) if and only if ηa is a subgroup of µa for each a ≤ a0, (ii) η ∈ L(µ) if and only if η>a is a subgroup of µ> a for each a < a0 provided L is a chain. Below we extend a result from [1] to L-setting: Proposition 2. [? ] Let η, θ ∈ L(µ). Then, η ◦ θ ∈ L(µ) if and only if η ◦ θ = θ ◦ η . Proposition 3. Let η, θ ∈ L(µ). Then, η and θ ⊆ η ◦ θ if and only if η(e) = θ(e). If η ∈ Lµ and ⟨η⟩µ denotes the L-subgroup of µ generated by η, then it can be easily verified that ⟨η⟩µ = ⟨η⟩. We recall the definition of a normal L-subgroup of an L-group. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2091 Definition 6. [3] Let η ∈ L(µ). Then, η is said to be a normal L-subgroup of µ, if for all x, y ∈ G η(yxy−1) ≥ η(x) ∧ µ(y). The set of normal L-subgroup of µ is denoted by NL(µ). Theorem 3. Let η ∈ L(µ). Then, η ∈ NL(µ) if and only if ηa is a normal subgroup of µa for each a ≤ η(e). Theorem 4. Let f : G → K be a group homomorphism. If µ ∈ L(G) and ν ∈ L(K), then (i) f(η) is an L-subgroup of f(µ) for each η ∈ L(µ), (ii) f−1(θ) is an L-subgroup of f−1(ν) for each θ ∈ L(ν). Theorem 5. [3] Let η, θ ∈ L(µ). Then, (i) η ◦ θ ∈ L(µ) if η or θ ∈ NL(µ), (ii) η ◦ θ ∈ NL(µ) if η and θ ∈ NL(µ). Theorem 6. [? ] Let η ∈ L µ . Let a0 = ∨ x∈G {η (x)} and define an L-subset η̂ of G by η̂ (x) = ∨ a≤a0 {a : x ∈ ⟨ηa⟩}. Then, η̂ ∈ L(µ) and η̂ = ⟨η⟩. Corollary 1. Let η ∈ L µ . Then, η̂(e) = ∨ x∈G {η(x)}. Definition 7. [15] Let η ∈ Lµ. Then, η is said to have sup-property if for every non-empty subset A of G, there exists a0 ∈ A such that ∨ a∈A {η(a)} = η(a0). Theorem 7. [? ] Let η ∈ Lµ and possesses sup-property. Then, define an L-subset η̂ of G by η̂ (x) = ∨ a∈Imη {a : x ∈ ⟨ηa⟩}. Then, η̂ ∈ L(µ) and η̂ = ⟨η⟩. Moreover, η̂ possesses sup-property and Im η̂ ⊆ Im η. Recall the following from [3]: Definition 8. Let η, θ ∈ Lµ. Then, the commutator of η and θ is an L-subset (η, θ) of G defined as follows: (η, θ)(x) = { ∨{η(y) ∧ θ(z)} if x = [y, z] for some y, z ∈ G, inf η ∧ inf θ if x ̸= [y, z] for any y, z ∈ G. Clearly, (η, θ) ⊆ µ. The commutator L-subgroup of η and θ is defined as ⟨(η, θ)⟩ and we write ⟨(η, θ)⟩ = [η, θ]. Note that inf(η, θ) = inf η ∧ inf θ and [η, θ] ∈ L(µ). N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2092 Here recall that if η is non-constant and η ̸= µ, then η is said to be a proper L-subgroup of µ. Also, η is said to be a trivial L-subgroup of µ if its chain of level subgroups contains only {e} and G. Clearly, η is a proper L-subgroup of µ if and only if η has distinct tip and tail, and η ̸= µ. Definition 9. Let η be a proper L-subgroup of µ. Then, we define an L-subgroup of µ contained in η, denoted by ηa0t0 , as follows: ηa0t0 (y) = { a0, if y = e, t0, if y ̸= e, where a0 = η(e) and t0 = inf η. Here ηa0t0 , a trivial L-subgroup of µ, is called the trivial L-subgroup of η. 3. Subnormal L-subgroup In order to introduce the concept of subnormality in the L-setting, we start with the definition of conjugate of an L-subset by an L-subset like its counterpart in classical group theory.Then, the notion of normal closure of an L-subgroup is defined and utilized to formulate the concept of subnormal L-subgroups. Definition 10. [4] Let η, θ ∈ Lµ. Define an L-subset θηθ−1 of G as follows: θηθ−1(x) = ∨ x=zyz−1 {η(y) ∧ θ(z)} for each x ∈ G. We call θηθ−1 the conjugate of η by θ. Clearly, θηθ−1 ⊆ µ. Hence the L-subgroup ⟨θηθ−1⟩ ∈ L(µ) and is denoted by ηθ. The following lemma is instrumental in the development of this paper: Lemma 1. Let η, θ ∈ L(µ). Then, (i) η ⊆ θηθ−1 provided η(e) ⩽ θ(e), (ii) ∨ x∈G {θηθ−1(x)} = θηθ−1(e) = η(e) ∧ θ(e), (iii) θηθ−1(x) = θηθ−1(x−1) for each x ∈ G, (iv) θηθ−1(gxg−1) ≥ θηθ−1(x) ∧ θ(g) for each x, g ∈ G. Proof. The proofs of (i)-(iii) are obvious. We only prove (iv). So, let x, g ∈ G. Then, θηθ−1(x) ∧ θ(g) = { ∨ x=zyz−1 {η(y) ∧ θ(z)} } ∧ θ(g) N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2093 = ∨ x=zyz−1 {η(y) ∧ θ(z) ∧ θ(g)} (as L is a completely distributive lattice) ≤ ∨ x=zyz−1 gxg−1=(gz)y(gz)−1 {η(y) ∧ θ(gz)} = θηθ−1(gxg−1). Corollary 2. Let η ∈ L(µ). Then, η ⊆ µηµ−1 ⊆ µ. Our next definition leads to the notion of normal closure of an L-subgroup of an L- group which in its essence is parallel to classical group theory. Definition 11. Let η ∈ L(µ). Then, the normal closure of η in µ is defined as the L-subgroup of µ generated by the conjugate ‘µηµ−1’. It is denoted by ηµ. Thus, ηµ = ⟨µηµ−1⟩. Note that by Corollary 2, the normal closure ηµ is an L-subgroup of µ containing η. In classical group theory, the normal closure of a subgroup of a group G is the smallest normal subgroup of G which contains the given subgroup. For the sake of completeness, recall the following from [2] : Theorem 8. Let η ∈ L(µ). Then, the normal closure ηµ is the least normal L-subgroup of µ containing η. Proof. As ηµ is an L-subgroup of µ containing η, we first establish that ηµ ∈ NL(µ). For this purpose, let x, g ∈ G. Note that, by Lemma 1,∨ y∈G µηµ−1(y) = η(e). Hence in view of Theorem 6, ηµ(x) ∧ µ(g) = { ∨ a≤η(e) {a : x ∈ ⟨(µηµ−1)a⟩} } ∧ µ(g) = ∨ a≤η(e) {a ∧ µ(g) : x ∈ ⟨(µηµ−1)a⟩}. (As L is a completely distributive lattice) Next, claim that if x ∈ ⟨(µηµ−1)a⟩, then ηµ(gxg−1) ≥ a ∧ µ(g). N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2094 In order to prove the claim, let x ∈ ⟨(µηµ−1)a⟩. Then, x = x1 x2 · · · xn, where either xi or x−1 i ∈ (µηµ−1)a for each i. Thus, gxg−1 = (gx1g −1)(gx2g −1) · · · (gxng−1). By Lemma 1, for each i µηµ−1(gxig −1) ≥ µηµ−1(xi) ∧ µ(g). Again by Lemma 1, for each i µηµ−1(xi) = µηµ−1(x−1 i ). As xi or x −1 i ∈ (µηµ−1)a, it follows that µηµ−1(gxig −1) ≥ a ∧ µ(g) for each i. Hence, n∧ i=1 {µηµ−1(gxig −1)} ≥ a ∧ µ(g). Therefore, ηµ(gxg−1) ≥ n∧ i=0 {ηµ(gxig−1)} ≥ n∧ i=1 {µηµ−1(gxig −1)} ≥ a ∧ µ(g). This establishes the claim. Consequently, ηµ(x) ∧ µ(g) = ∨ a≤η(e) {a ∧ µ(g) : x ∈ ⟨(µηµ−1)a⟩} ≤ ∨ a≤η(e) {ηµ(gxg−1) : x ∈ ⟨(µηµ−1)a⟩} = ηµ(gxg−1). Lastly to prove that ηµ is the least normal L-subgroup of µ containing η, let λ be a normal L-subgroup of µ containing η and x ∈ G. Then, µηµ−1(x) = ∨ x=zyz−1 {η(y) ∧ µ(z)} ≤ ∨ x=zyz−1 {λ(y) ∧ µ(z)} (as η ⊆ λ) N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2095 ≤ ∨ x=zyz−1 {λ(zyz−1)} (as η ∈ NL(µ)) = λ(x). Thus µηµ−1 ⊆ λ and as λ is an L-subgroup of µ, we have ηµ = ⟨µηµ−1⟩ ⊆ λ. This proves our result. Corollary 3. Let η ∈ L(µ). Then, η ∈ NL(µ) if and only if ηµ = η. Moreover, Proposition 4. Let η ∈ L(µ). Then, ηµ(e) = η(e). Next, we provide an example of the ‘normal closure’ of an L-subgroup of an L-group: Example 1. Let D8 = {⟨x, y⟩ : x2 = e = y8;xy = y−1x} be the dihedral group of degree 8. If D4 = {⟨x, y2⟩ : x2 = e = (y2)4;xy−2 = y−2x}, is a dihedral subgroup of D8, then define the following L-subsets of D8: µ(z) =  1 2 if z ∈ D4, 1 4 if z ∈ D8 ∼ D4; η(z) =  1 3 if z ∈ ⟨x⟩, 1 6 if z ∈ D4 ∼ ⟨x⟩, 1 9 if z ∈ D8 ∼ D4. Here A ∼ B means usual set difference and ⟨x⟩ denotes subgroup of D8 generated by ‘x’. Clearly η ⊆ µ, η ̸= µ and η, µ ∈ L(G). It can be verified easily that the conjugate ‘µηµ−1’ is defined by the following level subsets: (µηµ−1) 1 3 = {e, x, xy4}, (µηµ−1) 1 4 = {e, x, xy2, xy4, xy6}, (µηµ−1) 1 6 = {e, x, xy2, xy4, xy6, y2, y4, y6}and (µηµ−1) 1 9 = D8. Consequently, if K4 = {e, x, y4, xy4} is a Klein-4 subgroup of D8, then by applying the construction of Theorem 6 on the L-subset ‘η’ of µ and after doing necessary calculations, N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2096 we get L-subgroup ‘ηµ = ⟨µηµ−1⟩’ as follows: ηµ(z) =  1 3 if z ∈ K4, 1 4 if z ∈ D4 ∼ K4, 1 9 if z ∈ D8 ∼ D4. In the following, we discuss some properties of conjugate of L-subsets where the L- subsets in question are L-subgroups of a given parent L-group. The significance of such properties have already been shown in classical group theory for establishing certain prop- erties of ith normal closure of a subgroup of a group. Theorem 9. Let η, θ, γ ∈ L(µ). Let θ ⊆ γ and η(e) = θ(e). Then, (i) (ηγ)θ = ηγ, (ii) ( ηθ )γ = ηγ, (iii) ηθ◦η = ηθ. Proof. (i) In view of Lemma 1 and Corollary 1 and using the fact θ ⊆ γ, we have ηγ(e) = η(e) ∧ γ(e) = θ(e). Hence by Lemma 1, ηγ ⊆ (ηγ)θ. In order to prove the reverse inclusion, firstly we shall show that θηγθ−1 ⊆ ηγ . So let x ∈ G. Then, θηγθ−1(x) = ∨ x=yiziy −1 i {ηγ(zi) ∧ θ(yi)}. Here, in view of Lemma 1 and Corollary 1, we observe that∨ x∈G {γηγ−1(x)} = ηγ(e) = θ(e) = η(e). Thus γηγ−1(e) = η(e). Therefore, in view of Theorem 6, as ηγ = ⟨γηγ−1⟩, we have ηγ(zi) = ∨ aij≤η(e) {aij : zi ∈ ⟨(γηγ−1)aij ⟩}. Therefore, θηγθ−1(x) = ∨ x=yiziy −1 i {ηγ(zi) ∧ θ(yi)} N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2097 = ∨ x=yiziy −1 i { ∨ aij≤η(e) {aij : zi ∈ ⟨ ( γηγ−1 ) aij ⟩ } ∧ θ(yi)} = ∨ x=yiziy −1 i { ∨ aij≤η(e) {aij ∧ θ(yi) : zi ∈ ⟨ ( γηγ−1 ) aij ⟩} } (as L is a completely distributive lattice) = ∨ x=yiziy −1 i ai j ≤η(e) {aij ∧ θ(yi) : zi ∈ ⟨ ( γηγ−1 ) aij ⟩} ≤ ∨ x=yiziy −1 i ai j ≤η(e) {aij ∧ γ(yi) : zi ∈ ⟨ ( γηγ−1 ) aij ⟩}. (As θ ⊆ γ) Now, let yi ∈ G be such that x = yiziy −1 i for some zi ∈ ⟨ ( γηγ−1 ) aij ⟩ and aij ≤ η(e). We write caij ,yi = aij ∧ γ(yi). Then, caij ,yi ≤ η(e). We shall establish that yiziy −1 i ∈ ⟨ ( γηγ−1 ) c ai j ,yi ⟩. As zi ∈ ⟨ ( γηγ−1 ) aij ⟩, we have, zi = zi1z i 2...z i n where either zik or (zik) −1 ∈ (γηγ−1)aij for k = 1, 2, ..., n. Thus, yiziy −1 i = (yiz i 1y −1 i )(yiz i 2y −1 i )...(yiz i ny −1 i ). By Lemma 1, for each k γηγ−1(yiz i ky −1 i ) ≥ γηγ−1(zik) ∧ γ(yi). Moreover, again by Lemma 1, γηγ−1(zik) = γηγ−1((zik) −1) and as zik or (z i k) −1 ∈ (γηγ−1)aij , it follows that γηγ−1(yiz i ky −1 i ) ≥ aij ∧ γ(yi) = caij ,yi . Hence yiz i ky −1 i ∈ (γηγ−1)c ai j ,yi . Again, in view of Lemma 1, γηγ−1(yi(z i k) −1y−1 i ) = γηγ−1(yiz i ky −1 i ). Then, using the above arguments, we have yi(z i k) −1y−1 i ∈ (γηγ−1)c ai j ,yi . Consequently, yi(z i k) −1y−1 i and yiz i ky −1 i ∈ (γηγ−1)c ai j ,yi . This implies yiziy −1 i ∈ ⟨(γηγ−1)c ai j ,yi ⟩. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2098 Thus our claim is established. Therefore, we have θηγθ−1(x) ≤ ∨ x=yiziy −1 i ai j ≤η(e) {aij ∧ γ(yi) : zi ∈ ⟨ ( γηγ−1 ) aij ⟩} ≤ ∨ x=yiziy −1 i ) c ai j ,yi ≤η(e) {caij ,yi : yiziy −1 i ∈ ⟨(γηγ−1)c ai j ,yi ⟩} ≤ ∨ x=yiziy −1 i b≤η(e) {b : yiziy−1 i ∈ ⟨(γηγ−1)b⟩} ≤ ∨ d≤η(e) {d : x ∈ ⟨(γηγ−1)d⟩} = ⟨γηγ−1⟩(x) (by Theorem 6) = ηγ(x). Consequently, θηγθ−1 ⊆ ηγ . As ηγ ∈ L(µ), it follows that ⟨θηγθ−1⟩ = (ηγ)θ ⊆ ηγ . This establishes the desired equality. (ii) As θ ⊆ γ, by the definition of a conjugate L-subgroup, we have ηθ ⊆ ηγ and hence (ηθ)γ ⊆ (ηγ)γ . By part(i), (ηγ)γ = ηγ . Thus, (ηθ)γ ⊆ ηγ . In order to prove the reverse in- clusion, note that η(e) = θ(e) so that, by Lemma 1, η ⊆ ηθ. Therefore, we have ηγ ⊆ (ηθ)γ . This establishes the desired equality. (iii) As η(e) = θ(e), by Proposition 3, we have θ ⊆ θ ◦ η . Thus by the definition of a conjugate L-subset, we get ηθ ⊆ ηθ◦η. Now to prove the reverse inclusion, we take λ = θ ◦ η. Firstly, we shall show that ληλ−1 ⊆ ηθ. So let x ∈ G and consider ληλ−1(x) = ∨ x=yiziy −1 i {η(zi) ∧ λ(yi)} = ∨ x=yiziy −1 i { η(zi) ∧ { ∨ y=ui jv i j {θ(uij) ∧ η(vij)} }} = ∨ x=yiziy −1 i { ∨ yi=ui jv i j {η(zi) ∧ {θ(uij) ∧ η(vij)}} } (as L is a completely distributive lattice) N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2099 = ∨ x=yiziy −1 i yi=ui j vi j {η(zi) ∧ θ(uij) ∧ η(vij)} ≤ ∨ x=ui j(v i jzi(v i j) −1)(ui j) −1 {η(vijzi(vij)−1) ∧ θ(uij)} ≤ θηθ−1(x) ≤ ηθ(x). This establishes the claim. As ηθ ∈ L(µ), it follows that ηθ◦η = ⟨ληλ−1⟩ ⊆ ηθ. This establishes the desired equality. Theorem 10. Let η, θ, γ ∈ L(µ) be such that γ(e) = η(e) = θ(e). Then, (ηθ)γ = ηγ◦θ. Proof. As γ(e) = θ(e), by Proposition 3, γ ⊆ γ ◦ θ. Hence, (ηθ)γ ⊆ (ηθ)γ◦θ. By Theorem 9, we have (ηθ)γ◦θ = ηγ◦θ. Therefore, (ηθ)γ ⊆ ηγ◦θ. For the reverse inclusion, we take λ = γ ◦ θ. We shall establish that ληλ−1 ⊆ γηθγ−1. So let x ∈ G and consider ληλ−1(x) = ∨ x=yiziy −1 i {η(zi) ∧ λ(yi)} = ∨ x=yiziy −1 i { η(zi) ∧ { ∨ yi=ui jv i j {γ(uij) ∧ θ(vij)} }} = ∨ x=yiziy −1 i { ∨ yi=ui jv i j {η(zi) ∧ {γ(uij) ∧ θ(vij)}} } (as L is a completely distributive lattice) = ∨ x=yiziy −1 i yi=ui j vi j {η(zi) ∧ γ(uij) ∧ θ(vij)} ≤ ∨ x=ui j(v i jzi(v i j) −1)(ui j) −1 {θηθ−1(vijzi(v i j) −1) ∧ γ(uij)} (by the definition of a conjugate L-subset) N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2100 ≤ ∨ x=ui j(v i jzi(v i j) −1)(ui j) −1 {ηθ(vijzi(vij)−1) ∧ γ(uij)} ≤ γηθγ−1(x). This establishes the claim and hence ηγ◦θ = ⟨ληλ−1⟩ ⊆ ⟨γηθγ−1⟩ = (ηθ)γ . This proves the result. Theorem 11. Let η, θ ∈ L(µ) and ηθ = η. Then, η ◦ θ ∈ L(µ). Proof. Firstly we establish that ηθ ◦ θ = θ ◦ ηθ. Note that, by Lemma 1, we have ∨ x∈G θηθ−1(x) = η(e)∧θ(e). Let x ∈ G and a0 = η(e)∧θ(e). Then, ηθ ◦ θ(x) = ∨ x=uivi {ηθ(ui) ∧ θ(vi)} = ∨ x=uivi {{ ∨ ai j ≤a0 {aij : ui ∈ ⟨ ( θηθ−1 ) aij ⟩} } ∧ θ(vi) } = ∨ x=uivi {{ ∨ aij≤a0 {aij ∧ θ(vi) : ui ∈ ⟨ ( θηθ−1 ) aij ⟩} } ∧ θ(vi) } . ( As L is a completely distributive lattice) Let aij ≤ a0 and ui ∈ ⟨(θηθ−1)aij ⟩. It can be verified, as in Theorem 9, that v−1 i uivi ∈ ⟨(θηθ−1)aij∧θ(vi) ⟩. Thus, ηθ ◦ θ(x) = ∨ x=uivi {{ ∨ aij≤a0 {aij ∧ θ(vi) : ui ∈ ⟨ ( θηθ−1 ) aij ⟩} } ∧ θ(vi) } ≤ ∨ x=uivi {{ ∨ c≤a0 {c : v−1 i uivi ∈ ⟨ ( θηθ−1 ) c ⟩ } ∧ θ(vi) } = ∨ x=vi(v −1 i uivi) {ηθ(v−1 i uivi) ∧ θ(vi)} (by Theorem 6) ≤ θ ◦ ηθ(x). N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2101 Similarly, we can prove that θ ◦ ηθ ⊆ ηθ ◦ θ. This implies θ ◦ ηθ = ηθ ◦ θ. As ηθ = η, we have η ◦ θ = θ ◦ η. Therefore, by Proposition 2, η ◦ θ ∈ L(µ). In order to introduce the notion of subnormality of an L-subgroup of an L-group, firstly we define a descending series. For η ∈ L(µ), define a series of L-subgroups of µ inductively as follows: η0 = µ, η1 = ηµ, η2 = ηη1 , . . . , ηi = ηηi−1 . . . By Theorem 8, η1 is the smallest normal L-subgroup of µ containing η and η2 is the smallest normal L-subgroup of η1 containing η and so on. Thus, we have η ⊆ · · ·◁ ηi+1 ◁ ηi ◁ · · ·◁ η1 ◁ η0 = µ . This inductively defined series is known as the normal closure series of η in µ and we call ηi the ith normal closure of η in µ. In view of Proposition 4 and using the Principle of Mathematical Induction, we have: Proposition 5. Let η ∈ L(µ). Then, ηi(e) = η(e) for each i. Theorem 12. Let η, θ ∈ L(µ) such that η(e) = θ(e). Let ηi be the ith normal closure of η in µ and ηθi = ηi. Then, ηi+1 ∈ NL(ηi ◦ θ). Proof. In order to establish the result, firstly we show that ηi+1 ⊆ ηi ◦ θ. By Proposition 5, for each i, we have ηi(e) = η(e). And since η(e) = θ(e), we have ηi(e) = θ(e) for each i. Thus in view of Proposition 3, ηi ⊆ ηi ◦ θ and hence by the definition of ith normal closure of η in µ, we obtain ηi+1 ⊆ ηi ⊆ ηi ◦ θ. Moreover as ηθi = ηi, by Theorem 11, ηi ◦ θ ∈ L(µ). Thus, ηi+1 ∈ L(ηi ◦ θ). Here note that θηi+1θ −1(e) = ηi+1(e) ∧ θ(e) (by Lemma 1) = η(e) ∧ θ(e) (as ηi(e) = η(e)) = η(e) (as η(e) = θ(e)) = ηi+1(e). So by Lemma 1, ηi+1 ⊆ θηi+1θ −1. Now to prove that ηi+1 ∈ NL(ηi ◦ θ), let x, g ∈ G and consider ηi+1(x) ∧ ηi ◦ θ(g) ≤ θηi+1θ −1(x) ∧ ηi ◦ θ(g) =  ∨ x=yjzjy −1 j {ηi+1(zj) ∧ θ(yj)}  ∧ { ∨ g=ukvk {ηi(uk) ∧ θ(vk)} } N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2102 = ∨ x=yjzjy −1 j { {ηi+1(zj) ∧ θ(yj)} ∧ { ∨ g=ukvk {ηi(uk) ∧ θ(vk)} }} (as L is a completely distributive lattice) = ∨ x=yjzjy −1 j { ∨ g=ukvk {ηi+1(zj) ∧ θ(yj)} ∧ {ηi(uk) ∧ θ(vk)} } (as L is a completely distributive lattice) = ∨ x=yjzjy −1 j g=ukvk {ηi+1(zj) ∧ θ(yj) ∧ ηi(uk) ∧ θ(vk)}} ≤ ∨ x=yjzjy −1 j g=ukvk {ηi+1(zj) ∧ θ(vkyj) ∧ ηi(uk)} (as θ ∈ L(µ)) = ∨ gxg−1=ukvkyjzjy −1 j v−1 k u−1 k {ηi+1(zj) ∧ θ(vkyj) ∧ ηi(uk)} ≤ ∨ gxg−1=uk(vkyjzjy −1 j v−1 k )u−1 k { θηi+1θ −1(vkyjzjy −1 j v−1 k ) ∧ ηi(uk) } ≤ ∨ gxg−1=uk(vkyjzjy −1 j v−1 k )u−1 k { ηθi+1(vkyjzjy −1 j v−1 k ) ∧ ηi(uk) } = ∨ gxg−1=uk(vkyjzjy −1 j v−1 k )u−1 k { ηi+1(vkyjzjv −1 k y−1 j ) ∧ ηi(uk) } (as ηθi+1 = ηi+1) ≤ ∨ gxg−1=uk(vkyjzjy −1 j v−1 k )u−1 k { ηi+1(ukyjvkzjv −1 k y−1 j u−1 k ) } (as ηi+1 ∈ NL(ηi)) = ηi+1(gxg −1). Now, define the notion of subnormal L-subgroup of an L-group as follows: Definition 12. Let η ∈ L(µ) and ηi be the ith normal closure of η in µ. If there exists a non negative integer m such that ηm = η ◁ ηm−1 ◁ · · ·◁ η0 = µ, then η is known as a subnormal L-subgroup of µ with defect m. We shall denote a sub- normal L-subgroup η of µ with defect m by η m ◁ µ. If η is a subnormal L-subgroup of µ, then we shall also write ’η is subnormal in µ.’ Remark 1. Obviously m equals 0 if η = µ and m = 1 if η ∈ NL(µ) and η ̸= µ. The following example illustrates the notion of subnormal L-subgroup of an L-group: N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2103 Example 2. Recall that in Example 1, the normal closure ‘ηµ’ of η in µ is an L-subgroup of µ defined as follows: η1(z) = ηµ(z) =  1 3 if z ∈ K4, 1 4 if z ∈ D4 ∼ K4, 1 9 if z ∈ D8 ∼ D4. Similarly, one can verify that the conjugate ‘η1η(η1) −1’ is defined by the following level subsets: (η1η(η1) −1) 1 3 = {e, x, xy4}, (η1η(η1) −1) 1 6 = {e, x, xy2, xy4, xy6, y2, y4, y6}, (η1η(η1) −1) 1 9 = D8 . So in view of Theorem 6, the normal closure of η in η1 has the following definition: η2(z) = ⟨η1η(η1)−1⟩(z) =  1 3 if z ∈ K4, 1 6 if z ∈ D4 ∼ K4, 1 9 if z ∈ D8 ∼ D4. Moreover, one can verify that the conjugate ‘η2η(η2) −1’ is defined by the following level subsets: (η2η(η2) −1) 1 3 = ⟨x⟩, (η2η(η2) −1) 1 6 = D4, and (η2η(η2) −1) 1 9 = D8. As η2η(η2) −1 ∈ L(µ), we have η3 = ηη2 = η2η(η2) −1 = η. Consequently, we have η3 = η ◁ η2 ◁ η1 ◁ η0 = µ. Hence η is a subnormal L-subgroup of µ with defect 3. Next, we provide the definition of a subnormal seies for an L-subgroup: Definition 13. Let η ∈ L(µ). A finite series θ0 = µ, θ1, θ2, ..., θm = η of L-subgroups of µ such that η = θm ◁ θm−1 ◁ ... ◁ θ0 = µ is said to be a subnormal series of η. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2104 We shall describe the notion of a subnormal L-subgroup through the notion of above defined subnormal series like their classical counterparts. The following lemma, is a step in this direction: Lemma 2. Let η ∈ L(µ) and η ⊆ · · ·◁ ηi+1 ◁ ηi ◁ · · ·◁ η1 ◁ η0 = µ be the normal closure series of η. If there exists a descending series γ0 = µ, γ1, ..., γi... of L-subgroups of µ such that η ⊆ ... ◁ γi+1 ◁ ...γ1 ◁ γ0 = µ, then ηi ⊆ γi. Proof. We shall prove the result by induction on i. For i = 1, γ1 is a normal L- subgroup of µ containing η. But by Theorem 8, η1 = ηµ is the smallest normal subgroup of µ containing η. Hence η1 ⊆ γ1. Now, we set the induction hypothesis as ηk ⊆ γk. Thus, in view of the definition of a conjugate L-subset, we have ηk+1 = ηηk ⊆ ηγk . (1) Note that as η ⊆ γk, by Theorem 8, ηγk is the smallest normal L-subgroup of γk containing η. Moreover, γk+1 is also a normal L-subgroup of γk containing η. This implies ηγk ⊆ γk+1. Therefore, by using (1), we have ηk+1 = ηηk ⊆ ηγk ⊆ γk+1. This proves the result completely. The following result inter-connects the notions of subnormality and subnormal L-series of an L-subgroup: Theorem 13. Let η ∈ L(µ). Then, η is a subnormal L-subgroup of µ having defect m if and only if η has a subnormal series η = γm ◁ ... ◁ γi+1 ◁ ...γ1 ◁ γ0 = µ, of length m and m is the smallest length of such a subnormal series. Proof. If η is a subnormal L-subgroup of µ, then the normal closure series is the required subnormal series. Conversely, suppose that η has a subnormal series say η = θm ◁ θm−1 ◁ ... ◁ θ0 = µ. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2105 Then, by Lemma 2, ηm ⊆ θm. As η = θm, we have η ⊆ ηm ⊆ θm = η. Hence, ηm = η. This proves that η is a subnormal L-subgroup of µ. The following results are established by using the above theorem: Theorem 14. Let η be a subnormal L-subgroup of µ with defect m. (i) Let θ ∈ L(µ). Then, η ∩ θ is a subnormal L-subgroup of θ with defect c where c ≤ m. In particular, η is a subnormal L-subgroup of λ where λ ∈ L(µ) such that η ⊆ λ with defect c where c ≤ m, (ii) Let θ ∈ NL(µ). Then, η ◦ θ is a subnormal L-subgroup of µ with defect c where c ≤ m. It can be seen easily that the intersection of any finite set of subnormal L-subgroups is again subnormal. More generally : Theorem 15. Let {θi : i ∈ I} be a family of subnormal L-subgroups such that defect of θi is mi where mi ≤ m. Then, ⋂ i∈I θi is a subnormal L-subgroup of µ with defect c where c ≤ m. Next result determines the transitivity of the notion of subnormality. Theorem 16. Let η, θ ∈ L(µ) such that η is a subnormal L-subgroup of θ with defect m and θ is a subnormal L-subgroup of µ with defect n. Then, η is a subnormal L-subgroup of µ with defect m+n. The following theorem establishes that the subnormality in L-setting is also preserved under the action of a homomorphism and its inverse image: Theorem 17. Let η ∈ L(µ) and f : G → K be a group homomorphism. Then, (i) if η is a subnormal L-subgroup of µ with defect n, then f(η) is a subnormal L-subgroup of f(µ) with defect m where m ≤ n, (ii) if η is a subnormal L-subgroup of µ with defect n, then f−1(η) is a subnormal L-subgroup of f−1(µ) with defect m where m ≤ n, provided that the group homo- moirphism f is onto. 4. Subnormal L-subgroups and nilpotency In this section, we characterize subnormal L-subgroups by the usual group theoretic subnormality of the level subsets of the given L-subgroups. We shall refer this as a level subset characterization of subnormality. Then, this characterization is used to establish that when the lattice L is an upper well ordered chain, then every L-subgroup of a nilpotent L-group is subnormal. For this purpose, we need to develop a nececcary mechanism. So we start with a characterization of the notion of sup-property which lends itself more easily for applications. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2106 Definition 14. A non-empty subset X of a lattice L is said to be a supstar subset of L if every non-empty subset A of X contains its supremum. That is, if supA = a0, then a0 ∈ A. Clearly, a subset of a supstar subset is again supstar. Proposition 6. Let η ∈ Lµ. Then, η has sup-property if and only if Imη is a supstar subset of L. The above characterization also allows a generalization of the concept of sup-property to an arbitrary family of L-subsets and hence widens the scope of its applications. Definition 15. Let {ηi}i∈I ⊆ Lµ. Then, {ηi}i∈I is said to be a supstar family if ∪ i∈I Imηi is a supstar subset of L. As a particular case, we say that two L-subsets θ and η are jointly supstar if Imθ ∪ Imη is a supstar subset of L. In view of the above definition and Proposition 6, we have the following: Proposition 7. Each member of a supstar family of L-subsets satisfies sup-property. Lemma 3. Let η ∈ Lµ and η has sup-property. If a0 = ∨ x∈G {η(x)}, then ⟨ηb⟩ = ⟨η⟩b for each b ≤ a0. Lemma 4. Let η ∈ L(µ) be such that µ and η are jointly supstar.. Then, Imη ∪ Imµ is a chain. Next, we prove: Lemma 5. Let η ∈ L(µ) be such that µ and η are jointly supstar. Then, Im ηµ ⊆ Im (µηµ−1) ⊆ Imµ ∪ Imη. Proof. Let a ∈ Im(µηµ−1). So, there exists x ∈ G such that a = (µηµ−1)(x). Now, define the following subset of G×G : C(x) = {(v, u) ∈ G×G : x = uvu−1}. Thus, a = (µηµ−1)(x) = ∨ (y,z)∈C(x) {η(y) ∧ µ(z)}. (1) Now, for any (y, z) ∈ C(x), {η(y), µ(z)} ⊆ Imη ∪ Imµ. As µ and η are jointly supstar, by Lemma 4, Imη ∪ Imη is a chain. Hence η(y) ∧ µ(z) = η(y) or µ(z) which is an element of Imη ∪ Imµ. As Imη ∪ Imµ is supstar, the subset N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2107 {η(y) ∧ µ(z) : (y, z) ∈ C(x)} ⊆ Imη ∪ Imµ contains its supremum a and hence a = (µηµ−1)(x) ∈ Imµ ∪ Imη. Therefore, Im (µηµ−1) ⊆ Imµ ∪ Imη. Further, as a subset of a supstar subset is again supstar, it follows that Im(µηµ−1) is also supstar. Hence by Proposition 6, µηµ−1 possesses sup-property. Hence in view of Theorem 7, it follows that Imηµ = Im⟨µηµ−1⟩ ⊆ Im (µηµ−1) ⊆ Imµ ∪ Imη. More generally we have: Lemma 6. Let η ∈ L(µ) be such that µ and η are jointly supstar. Then, for each i Im ηi+1 ⊆ Im (ηiηη −1 i ) ⊆ Imµ ∪ Imη, where ηi is the ith normal closure of η in µ. Corollary 4. Let η ∈ L(µ) be such that µ and η are jointly supstar. Then for each i, η and ηi are jointly supstar. Remark 2. Note that if η ∈ L(µ) and µ and η be jointly supstar, then the L-subset ηiηη −1 i possesses sup-property for each i. Below, we discuss the level subset of the normal closure of η in µ. Lemma 7. Let η ∈ L(µ). Then, (i) (ηµ)a = (ηa) µa for each a ≤ η(e) provided µ and η are supstar, (ii) (ηµ)>a = (η>a ) µ> a provided L is a chain. Proof. (i) Let a ≤ η(e). We show that (µηµ−1)a ⊆ µaηa(µa) −1. (1) So, let x ∈ (µηµ−1)a. Then, µηµ−1(x) = ∨ x=zyz−1 {η(y) ∧ µ(z)} ≥ a. As µ and η are jointly supstar, it can be verified, as in Lemma 5, that there exists y0 and z0 ∈ G such that x = z0y0z −1 0 and∨ x=zyz−1 {η(y) ∧ µ(z)} = η(y0) ∧ µ(z0). N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2108 This implies µηµ−1(x) = η(y0) ∧ µ(z0) ≥ a. Therefore, η(y0) and µ(z0) ≥ a. So, we have y0 ∈ ηa and z0 ∈ µa. Hence, x = z0y0z −1 0 ∈ µaηa(µa) −1. This establishes (1). In order to prove the reverse inclusion, let x ∈ µaηa(µa) −1. Then, there exist y0 ∈ ηa and z0 ∈ µa such that x = z0y0(z0) −1. Therefore, µηµ−1(x) = ∨ x=zyz−1 {η(y) ∧ µ(z)} ≥ η(y0) ∧ µ(z0) ≥ a. Hence x ∈ (µηµ−1)a. Thus, µaηa(µa) −1 ⊆ (µηµ−1)a. Therefore, µaηa(µa) −1 = (µηµ−1)a. Further, as µ and η are jointly supstar, by Remark 2 the L-subset µηµ−1 possesses sup- property. Hence, by Lemma 3, we have (ηµ)a = ⟨µηµ−1⟩a = ⟨(µηµ−1)a⟩ = ⟨µaηa(µa) −1⟩ = (ηa) µa . This proves (i). (ii) It can be prove by using the arguments of part(i). Let ηi be the ith normal closure of η in µ. Then, more generally we have : Lemma 8. Let η ∈ L(µ). Then, (i) (ηi)a = (ηa)i for each a ≤ η(e) provided µ and η are supstar, (ii) (ηi) > a = (η>a )i provided L is a chain. Theorem 18. Let η ∈ L(µ) be such that η and µ are jointly supstar. Then, η is subnormal having defect at most n if and only if each level subset ηa is subnormal having defect atmost n where a ≤ η(e). Proof. (Condition is necessary.) Since η is subnormal in µ having defect atmost n, by the definition, there exists a positive integer m ≤ n such that µ = η0 ⊇ η1 ⊇ · · · ⊇ ηm = η, (1) where ηi is the ith normal closure of η in µ. Let a ≤ η(e). As η ∈ L(µ), by Theorem 2, ηa is a subgroup of µa. We show that ηa is subnormal in µa. Note that, in view of (1) we have µa = (η0)a ⊇ (η1)a ⊇ · · · ⊇ (ηm)a = ηa. Further, as η and µ are jointly supstar, by Lemma 8 N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2109 (ηa)i = (ηi)a for all i. Thus, we have µa = (ηa)0 ⊇ (ηa)1 ⊇ · · · ⊇ (ηa)m = ηa. Consequently, the normal closure series of ηa in µa has the length atmost m. As m ≤ n, ηa is subnormal in µa having defect atmost n. (Condition is Sufficient.) In order to show that η is subnormal having defect atmost n, we construct the normal closure series of η in µ and show that its length is atmost n. Now, by the hypothesis, each non-empty level subgroup of η is subnormal having defect atmost n. Let maj be the defect of the level subgroup ηaj in µaj for any aj ≤ η(e). Then, the normal closure series of ηaj in µaj is given by µaj = (ηaj )0 ⊇ (ηaj )1 ⊇ ... ⊇ (ηaj )maj = ηaj for all aj ≤ η(e), (2) where (ηaj )i is the ith normal closure of ηaj in µaj . Next, let C = Imη ∪ {a ∈ Imµ : a ≤ η(e)}. Then, by Lemma 4, C is a chain. Let S = {maj ≤ n : aj ∈ C}. Then, supS ≤ n. We write m = supS ≤ n. Now, we shall construct the normal closure series of η in µ having the length m. In view of (2), we have µaj = (ηaj )0 ⊇ (ηaj )1 ⊇ ... ⊇ (ηaj )maj = ηaj for all aj ∈ C. Further, as maj ≤ m for each aj ∈ C, we insert m −maj times the level subgroup ηaj in the normal closure series of ηaj in µaj . Consequently, the normal closure series of ηaj in µaj aquire the length m. Obviously, for all l = 1, 2, ...,m−maj (ηaj )maj+l = ηaj . Thus µaj = (ηaj )0 ⊇ (ηaj )1 ⊇ ... ⊇ (ηaj )maj = ηaj for all aj ∈ C. (3) In view of Lemma 8, we have µaj = (η0)aj ⊇ (η1)aj ⊇ ... ⊇ (ηm)aj = ηaj for all aj ∈ C. Let aj , ak ∈ C and aj ≥ ak. Hence for each fixed i, it follows that (ηaj )i = (ηaj ) (ηaj )i−1 (by the definition of classical ith normal closure) = ⟨(ηaj )i−1ηaj ((ηaj )i−1) −1⟩ ⊆ ⟨(ηak)i−1ηak((ηak)i−1) −1⟩ = (ηak)i. (4) N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2110 Thus for a fixed i the ith normal closures of the level subsets ηaj for each aj ∈ C constitutes a chain. In view of the fact that the level subsets ηaj are ordinary subgroups, η ηaj aj = ηaj , insertion of the level subgroup ηaj in the chain of its ith normal closures does not affect the above containment. In view of Lemma 8 and by (4), we obtain (ηi)aj ⊆ (ηi)ak . In view of (3), for each i, we have (ηi−1)aj ⊇ (ηi)aj for each aj ∈ C. Now we claim that (ηi−1)aj ⊇ (ηi)aj for each aj ∈ Im µ ∪ Im η. Here observe that, in view of Proposition 5, the tip of ηi, for each i, is η(e). Hence (ηi)aj = ϕ for all aj ∈ {a ∈ Im µ : a ≰ η(e)}. This establishes the claim. Further, by Lemma 6, for each i Im ηi ⊆ Im η ∪ Im µ. Thus, in view of Proposition 1, we have ηi−1 ⊇ ηi for each i. Consequently, µ = η0 ⊇ η1 ⊇ · · · ⊇ ηm = η. Then, the normal closure series of η in µ has the length atmost m ≤ n. Therefore, η is a subnormal L-subgroup of µ having defect atmost n. Corollary 5. Let G be a group and H be its subgroup. Then, H is a subnormal subgroup of G if and only if 1H is a subnormal L-subgroup of 1G. The strong level subset characterization of subnormal L-subgroup can be obtained easily by using the arguments of Theorem 18. So, we state it without proof: Theorem 19. Let L be chain and η ∈ L(µ). Then, η is subnormal having defect at most n if and only if each strong level subset η>a is subnormal having defect atmost n where a < η(e). Recall the following result from classical group theory [11, 13]: N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2111 Theorem 20. Every subgroup of a nilpotent group is subnormal. Now, we shall deal with this result in L-group theory provided L is an upper well ordered chain. A chain is said to be upper well ordered if every non-empty subset of the given chain has a greatest element. Clearly, every subset of an upper well ordred chain is a supstar subset. Consequently, each L-subset for an upper well ordered chain L satisfies sup-property. Proposition 8. Let L be an upper well ordered chain and η, θ ∈ Lµ. Then, η and θ are jointly supstar if and only if η and θ possess sup-property. Below, we recall the definition of descending central chain of an L-subgroup η of µ from [3]: Take γ0(η) = η, γ1(η) = [γ0(η), η]. And in general, for each i, we define γi(η) = [γi−1(η), η]. Proposition 9. Let η ∈ L(µ). Then for each i, γi(η) ⊆ γi−1(η). Definition 16. Let η ∈ L(µ). Then, the chain η = γ0(η) ⊇ γ1(η) ⊇ · · · ⊇ γi(η) ⊇ · · · of L-subgroups of µ is called the descending central chain of η. Definition 17. Let η ∈ L(µ) with tip a0 and tail t0 and a0 ̸= t0. If the descending central chain η = γ0(η) ⊇ γ1(η) ⊇ · · · ⊇ γi(η) ⊇ · · · terminates finitely to the trivial L-subgroup ηa0t0 , then η is known as a nilpotent L-sub- group of µ. More precisely, η is said to be nilpotent of class c if c is the least non-negative integer such that γc(η) = ηa0t0 . In this case, the series η = γ0(η) ⊇ γ1(η) ⊇ · · · ⊇ γc(η) = ηa0t0 , is called the descending central series of η. If it is a nilpotent L-subgroup of µ, then we simply write η is nilpotent. Theorem 21. Let η ∈ L(µ) and possesses sup-property. Then, η is a nilpotent L-subgroup of µ of nilpotent length at most n if and only if each level subgroup ηa is a nilpotent subgroup of µa of nilpotent length at most n for each a ̸≤ inf η and a ≤ η(e). Theorem 22. Let η ∈ L(µ) and L be a chain. Then, η is a nilpotent L-subgroup of µ of nilpotent length at most n if and only if η>a is a nilpotent subgroup of µ> a of nilpotent length at most n, where inf η ≤ a < η(e). The notion of nilpotency of L-subgroup is exihibited in the following: N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2112 Example 3. Let G = Q8 and the evaluation lattice be given by the diagram : u f ba d Fig. 1 Consider the parent L-subgroup of G given by: µ(x) = { u if x ∈ C, d if x ∈ G \ C. Now define L-subset η of µ as given below: η(x) =  u if x ∈ C, d if x ∈ H1 \ C, a if x ∈ H2 \ C, b if x ∈ H3 \ C; where C = {±1}, H1 = {±1,±i}, H2 = {±1,±j}, H3 = {±1,±k}. Since the level subsets of η are normal subgroups of G, η is a normal L-subgroup of G and hence of µ. Now that η is a nilpotent L-subgroup of µ in view of Definition 17, we demonstrate this as follows: Note that G′ = {1,−1}. In order to obtain the members of descending central series of η, we set γ0(η) = η and consider the commutator (η, η) (η, η) (x) =  u if x = 1, d if x ∈ C \ {1}, f if x ∈ G \ C. N. Ajmal, I. Jahan., B. Davvaz / Eur. J. Pure Appl. Math, 15 (4) (2022), 2086-2115 2113 As the level subsets of (η, η) are subgroups of G, γ1(η) = [η, η] = (η, η). Next, we calculate the commutator : ((η, η) , η) (x) = { u if x = 1, f if x ∈ G \ {1}. Again, by the reasons as given above γ2(η) = [[η, η] , η] = ((η, η) , η). Observe that γ2(η) is not only an L-subgroup, it is the trivial L-subgroup of η and so the descending central series terminates at γ2(η), i. e. η = γ0(η) ⊇ γ1(η) ⊇ γ2(η) = ηuf . Consequently, η is a nilpotent L-subgroup of µ having nilpotent length 2. Theorem 23. Let L be an upper well ordered chain. Let η ∈ L(µ) and η be nilpotent having the tip a0 and the tail t0. If θ ∈ L(η) having the tail t0, then θ is a subnormal L-subgroup of η. Proof. Let θ ∈ L(η) having the tail t0. We shall show that the level subgroup θa is a subnormal L- subgroup of ηa for each a ≤ θ(e). If a ≤ t0, then obviously θa is a subnormal subgroup of ηa. If So, let t0 < a ≤ θ(e). As L is upper well ordered, it follows the L-subset η possesses sup-property. Now, as t0 < a ≤ θ(e) ≤ η(e) and η is nilpotent, by Theorem 21, the level subgroup ηa is nilpotent. Thus θa is a subgroup of the nilpotent subgroup ηa, Hence by Theorem 20, θa is a subnormal subgroup of ηa. Again as L is an upper well ordered chain, by using the arguments as above, it follows that both the L-subgroups η and θ possess sup-property. Therefore, by Proposition 8, η and θ are joinly supstar. Consequently, by Theorem 18, θ is a subnormal L-subgroup of η. In the part II of this paper, we develop a mechanism in order to tackle the join problem for subnormal L-subgroup. Theorem 24. Let η and θ be subnormal L-subgroups of µ . Let η(e) = θ(e) and η ◦ θ ∈ L(µ). Then, the following are equivalent: (i) η ◦ θ is subnormal in µ, (ii) ηθ is subnormal in µ, (iii) [η, θ] is subnormal in µ, REFERENCES 2114 5. Conclusion The Theory of L-groups is a very rich generalization of classical group theory. Here we study the group theoretic properties of posets of subgroups of a group or in particular chains of subgroups of a group rather than properties of a single subgroup. As an applica- tion of this work and the motivation for the development of L-group theory, we mention that if we replace the lattice L, in our work by the closed unit interval [0, 1], then we retrieve the corresponding version of fuzzy group theory. Moreover, as an application of this theory we also mention that if we replace the lattice L by the two elements set {0, 1}, then the results of classical group theory follow as simple corollaries of the corresponding results of L group theory. This way, L-group theory provides us a new language and a new tool for the study of the classical group theory. The classical group theory has been founded on abstract sets and therefore the language used for its development is formal set theory. On the other hand, L-group theory expresses itself through the language of functions. The functions which are lattice valued. 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