EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5908
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Jordan-Hölder Theorem under Intuitionistic Fuzzy
Groups

Paul Augustine Ejegwa1, Nasreen Kausar2, Tonguc Cagin3,∗

1 Department of Mathematics, Joseph Sarwuan Tarka University, P.M.B. 2373, Makurdi,
Nigeria
2 Department of Mathematics, Faculty of Arts and Sciences, Balikesir University,
10145 Balikesir, Türkiye
3 College of Business Administration, American University of the Middle East, Kuwait

Abstract. The theory of intuitionistic fuzzy groups is an algebraic structure derivable from the
utilization of groups in intuitionistic fuzzy sets. Many notions in group theory have been pre-
sented in intuitionistic fuzzy group theory. However, concepts like simple group, maximal normal
subgroup, normal series, composition series, and the Jordan-Hölder Theorem are open problems
in intuitionistic fuzzy group theory. Hence, this paper defines simple intuitionistic fuzzy groups,
maximal normal intuitionistic fuzzy subgroups, normal series for intuitionistic fuzzy groups, and
composition series for intuitionistic fuzzy groups with illustrations. In addition, the Jordan-Hölder
Theorem in intuitionistic fuzzy group theory is verified. It is shown that every intuitionistic fuzzy
group of a finite group possesses a composition series, and any two composition series for an
intuitionistic fuzzy group of a finite group are equivalent.

2020 Mathematics Subject Classifications: 03F55, 06D72, 08A72

Key Words and Phrases: Intuitionistic fuzzy groups, Intuitionistic fuzzy subgroups, simple
intuitionistic fuzzy groups, maximal normal intuitionistic fuzzy subgroups, normal series for intu-
itionistic fuzzy groups, composition series for intuitionistic fuzzy groups

1. Introduction

The introduction of fuzzy set theory (FST) helped in the resolution of uncertainty that
set theory could not handle by considering the membership degree of elements defined in
a unit closed interval, [0, 1]. The presentation of FST by Zadeh [1] has transformed
decision-making in real-life issues. Likewise, FST has prompted the study of fuzzy algebra
in line with the classical algebraic structures. Rosenfeld [2] introduced fuzzy groups as the
application of fuzzy sets to group theory and studied many group theoretic notions under

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5908

Email addresses: ejegwa.augustine@uam.edu.ng (P. A. Ejegwa),
nasreen.kausar@balikesir.edu.tr (N. Kausar), tonguc.cagin@aum.edu.kw (T. Cagin)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 2 of 11

fuzzy group setting. In addition, several group analogous concepts like fuzzy subgroups
and Frattini fuzzy subgroups were studied [3, 4].

Because of the drawback of the FST (because it considers only the membership degree
of elements), the idea of intuitionistic fuzzy sets (IFSs) was presented by Atanassov [5]
and numerous properties of IFSs have been presented [6, 7]. Biswas [8, 9] applied IFSs to
group theory by proposing intuitionistic fuzzy subgroups (IFSGs), Ahn [10] discussed the
several forms of sublattice of lattice of IFSGs, and proved the connection of the sublattices
of the lattice of IFGs. Certain properties of IFSGs were discussed in [11], Fathi and Salleh
[12] presented intuitionistic fuzzy groups (IFGs) based on intuitionistic fuzzy space and
discussed certain of their properties, and Yuan et al. [13] shared some additional insight
on IFSGs. Bal et al. [14] presented a note of kernel subgroups on IFGs and discussed
some properties of IFGs, Sharma [15] presented the direct product of IFSGs, and the
homomorphism of IFGs was discussed in [16].

In addition, the α and β cuts of IFGs were discussed in [17] and the t-IFSGs was
presented in [18]. Some important theorems of t-intuitionistic fuzzy isomorphism of t-
IFSGs and the fuzzification of the prominent Lagrange’s theorem were discussed in [19, 20].
Some algebraic descriptions of ε-IFSGs were discussed in [21], Shuaib et al. [22] discussed
the idea of η-IFSG using η-IFSs and showed that each IFSG is an η-IFSG. In addition,
the study defined η-intuitionistic fuzzy cosets and η-intuitionistic fuzzy normal subgroups
(IFNSGs) of a given group. Gulzar et al. [23] presented some properties of t-IFSGs, Rasuli
[24] discussed t- norn and s-norm of IFSGs, and Latif and Shuaib [25] applied t-IFSGs to
discuss the famous Sylow theory.

Furthermore, the notion of intuitionistic anti-fuzzy subgroups was deliberated in [26]
and Husban et al. [27] discussed complex intuitionistic fuzzy group (CIFG) by permitting
the degrees of membership and non-membership to include complex numbers. Subse-
quently, Husban et al. [28] discussed normality in CIFGs with some properties, Al-Sharoa
[29] presented α1,2 and β1,2 cuts of CIFSGs with their algebraic properties, and Rasuli [30]
discussed t- norn and s-norm for CIFSGs. Although many group theoretical properties
have been discussed in IFGs, it is certain that simple intuitionistic fuzzy groups, maximal
normal intuitionistic fuzzy subgroups, normal series for intuitionistic fuzzy groups, com-
position series for intuitionistic fuzzy groups, and the Jordan-Hölder Theorem under IFG
have not been studied. The motivation for this work is to establish simple groups, maximal
normal subgroups, normal series, composition series, and the Jordan-Hölder Theorem in
the context of IFGs. The following are the contributions of the article:

(i) The idea of simple group and maximal normality are established in the context of
IFGs.

(ii) Normal series and composition series for IFGs are presented, exemplied, and dis-
cussed with the aid of some theorems.

(iii) The famous Jordan-Hölder Theorem is investigated under IFG by using composition
series for IFGs.



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 3 of 11

The rest of the paper are outlined as follows: Section 2 presents IFSs, IFSGs, and
their properties; Section 3 introduces the concepts of simple IFGs, maximal normal IFSGs,
normal series for IFGs, composition series for IFGs, and the Jordan-Hölder Theorem under
IFGs; and Section 4 concludes the paper.

2. Preliminaries

Throughout the paper, the symbols S and G represent a non-empty set and a group,
respectively.

Definition 1 ([1]). A fuzzy subset ζ of S is presented as:

ζ = {⟨s, ζm(s)⟩ | s ∈ S}, (1)

where ζm : X → [0, 1] is the membership degree of s ∈ X.

Definition 2 ([2]). A fuzzy subset ζ of G is a fuzzy subgroup of G if

(i) ζm(xy) ≥ min
{
ζm(x), ζm(y)

}
∀ x, y ∈ G,

(ii) ζm(x−1) = ζm(x) ∀ x ∈ G.

In addition, ζm(e) = ζm(xx−1) ≥ min
{
ζm(x), ζm(x)

}
= ζm(x) ∀ x ∈ G, where e is the

unit element of G.

Definition 3 ([5]). An IFS λ of S is presented as:

λ = {⟨s, λm(s), λn(s)⟩ | s ∈ S}

= {⟨λm(s), λn(s)

s
⟩ | s ∈ S}

 , (2)

where λm : X → [0, 1] and λn : X → [0, 1] are the membership and non-membership grades
of s ∈ S and 0 ≤ λm(s) + λn(s) ≤ 1.

Definition 4 ([6]). Let λ and γ be IFSs of S. Then,

(i) λ = γ ⇐⇒ λm(s) = γm(s) and λn(s) = γn(s) ∀s ∈ S,

(ii) λ ⊆ γ ⇐⇒ λm(s) ≤ γm(s) and λn(s) ≥ γn(s) ∀s ∈ S,

(iii) λ ∩ γ =
{
⟨s,min{λm(s), γm(s)},max{λn(s), γn(s)}⟩ | s ∈ S

}
,

(iv) λ ∪ γ =
{
⟨s,max{λm(s), γm(s)},min{λn(s), γn(s)}⟩ | s ∈ S

}
.

Definition 5 ([8]). An IFS λ of G is an IFG/IFSG of G if

(i) λm(xy) ≥ min
{
λm(x), λm(y)

}
and λn(xy) ≤ max

{
λn(x), λn(y)

}
∀ x, y ∈ G,



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 4 of 11

(ii) λm(x−1) = λm(x) and λn(x
−1) = λn(x) ∀ x ∈ G.

In addition,

ζm(e) = λm(xx−1) ≥ min
{
λm(x), λm(x)

}
= λm(x),

λn(e) = λn(xx
−1) ≤ max

{
λn(x), λn(x)

}
= λn(x)

 , (3)

∀ x ∈ G, where e is the unit element of G.
The order of λ is defined by

|λ| =
k∑

i=1

λm(xi) +
k∑

i=1

λn(xi) ∀xi ∈ G. (4)

|λ| is defined by the finiteness of G or otherwise.

Definition 6 ([8]). Let λ and γ be IFGs of G. Then, λ is an IFSG of γ if λ ⊆ γ. Again,
λ is a proper IFSG of γ if λ ⊆ γ and β ̸= γ.

Definition 7 ([17]). Let λ be an IFG of G. Then, the support of λ defined by:

λ∗ = {x ∈ G | λm(x) > 0 and λn(x) < 0}, (5)

is a subgroup of G.

Definition 8 ([31]). Let λ and γ be IFGs of G. Then, the product λ ◦γ is an IFS defined
as:

(λ ◦ γ)(g) =


∨

x=yz min
{
λm(y), γm(z)

}
,
∧

x=yz max
{
λn(y), γn(z)

}
,

if ∃ x, y ∈ G such that x = yz
0, otherwise.

(6)

Definition 9 ([31]). An IFG λ of G is commutative if λm(xy) = λm(yx) and λn(xy) =
λn(yx) ∀x, y ∈ G. Succinctly, an IFG β is commutative if G is a commutative group.

Definition 10 ([31]). Let λ and γ be IFGs of G such that λ ⊆ γ. Then, λ is an intu-
itionistic fuzzy normal subgroup (IFNSG) of γ denoted as λ ◁ γ if λm(xy) = λm(yx) and
λn(xy) = λn(yx) ⇐⇒ λm(y) = λm(x−1yx) and λn(y) = λn(x

−1yx) ∀ x, y ∈ G.

Definition 11 ([32]). If λ is an IFSG of G. Then, yλ for y ∈ G defined by (yλ)m(x) =
λm(y−1x) and (yλ)n(x) = λn(y

−1x) ∀ x ∈ G is called the left intuitionistic fuzzy coset
(IFCS) of G. Similarly, λy for y ∈ G defined by (λy)m(x) = λm(xy−1) and (λy)n(x) =
λn(xy

−1) ∀ x ∈ G is called the right IFCS of G.

Definition 12 ([33]). Suppose λ and γ are IFGs of G and λ ◁ γ. Then, the collection of
the left/right IFCSs of λ such that xλ ◦ yλ = xyλ ∀ x, y ∈ G is called an intuitionistic
fuzzy factor group (IFFG) of γ by λ, represented by γ/λ.



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 5 of 11

3. Main Results

The Jordan-Hölder therorem has been discused in group [34] and the decomposition of
a crossed square [35]. The idea of the Jordan-Hölder therorem has not been investigated
under IFG. Before we introduce normal series for IFGs, composition series for IFGs, and
Jordan-Hölder therorem under IFGs, the notions of maximal IFNSG of a IFG and simple
IFGs are defined. Using Definition 10, we define a maximal IFNSG of an IFG as follows:

Definition 13. Let λ and γ be IFGs of G with λ ◁ γ. Then

(i) λ is a maximal non-trivial IFNSG if it is the largest proper non-trivial IFNSG of γ.

(ii) γ is simple if it has no proper non-trivial IFNSG.

Remark 1. An IFSG λ of an IFG γ in G is non-trivial if λ∗ = γ∗ = G and λ ̸= γ.
Otherwise, it is trivial.

Example 1. For a given cyclic group G = {1, a, a2, a3} where a4 = 1, a−1 = a3, (a2)−1 =
a2, and (a3)−1 = a, we define an IFG λ in G as:

λ = {⟨0.9, 0.05
1

⟩, ⟨0.5, 0.3
a

⟩, ⟨0.6, 0.2
a2

⟩, ⟨0.5, 0.3
a3

⟩}.

Certainly, λ is commutative IFG. Then, the following are the IFSGs of λ:

λ1 = {⟨0.4, 0.55
1

⟩, ⟨0.0, 0.8
a

⟩, ⟨0.1, 0.5
a2

⟩, ⟨0.0, 0.8
a3

⟩},

λ2 = {⟨0.5, 0.45
1

⟩, ⟨0.1, 0.7
a

⟩, ⟨0.2, 0.6
a2

⟩, ⟨0.1, 0.7
a3

⟩},

λ3 = {⟨0.6, 0.35
1

⟩, ⟨0.2, 0.6
a

⟩, ⟨0.3, 0.5
a2

⟩, ⟨0.2, 0.6
a3

⟩},

λ4 = {⟨0.7, 0.25
1

⟩, ⟨0.3, 0.5
a

⟩, ⟨0.4, 0.4
a2

⟩, ⟨0.3, 0.5
a3

⟩},

λ5 = {⟨0.8, 0.15
1

⟩, ⟨0.4, 0.4
a

⟩, ⟨0.5, 0.3
a2

⟩, ⟨0.4, 0.4
a3

⟩},

λ6 = {⟨0.9, 0.05
1

⟩, ⟨0.5, 0.3
a

⟩, ⟨0.6, 0.2
a2

⟩, ⟨0.5, 0.3
a3

⟩}.

Here, λ6 is a trivial IFSG of λ because λ6 = λ, and λi for i = 1, 2, 3, 4, 5 are proper
non-trivial IFNSG of λ, and so λ is not a simple IFG. In addition, λ5 is the maximal
non-trivial IFNSG of λ.

Example 2. In a symmetry group, Sk for k = 3, A3 = {ρ0, ρ1, ρ2} ⊆ S3 is a simple group
(ρ−1

1 = ρ2 and ρ−1
2 = ρ1). Then, an IFG of A3 is:

γ = {⟨0.6, 0.3
ρ0

⟩, ⟨0.4, 0.3
ρ1

⟩, ⟨0.4, 0.3
ρ2

⟩}.



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 6 of 11

Since ρ1.ρ2 = ρ2.ρ1 = ρ0, then γ is commutative. Then, the IFSGs of γ are:

γ1 = {⟨0.2, 0.7
ρ0

⟩, ⟨0.0, 0.7
ρ1

⟩, ⟨0.0, 0.7
ρ2

⟩},

γ2 = {⟨0.3, 0.6
ρ0

⟩, ⟨0.1, 0.6
ρ1

⟩, ⟨0.1, 0.6
ρ2

⟩},

γ3 = {⟨0.4, 0.5
ρ0

⟩, ⟨0.2, 0.5
ρ1

⟩, ⟨0.2, 0.5
ρ2

⟩},

γ4 = {⟨0.5, 0.4
ρ0

⟩, ⟨0.3, 0.4
ρ1

⟩, ⟨0.3, 0.4
ρ2

⟩},

γ5 = {⟨0.6, 0.3
ρ0

⟩, ⟨0.4, 0.3
ρ1

⟩, ⟨0.4, 0.3
ρ2

⟩}.

Here, γ5 is a trivial IFSG of γ because γ5 = γ, and γi for i = 1, 2, 3, 4 are proper non-trivial
IFNSG of γ, and so γ is not simple. In addition, γ4 is the maximal non-trivial IFNSG of
γ.

Example 3. Suppose we have an IFG δ = {⟨0.5,0.3ρ0
⟩, ⟨0.0,1ρ1

⟩, ⟨0.0,1ρ2
⟩} defined in A3 =

{ρ0, ρ1, ρ2} ⊆ S3. It is observed that δ has no proper non-trivial IFSG. Thus, δ is a simple
IFG.

Theorem 1. Every IFG which has no non-trivial IFSGs is simple.

Proof. The proof is simple from Example 3.

Definition 14. Let G be a finite group and λ be an IFG of G. Then, λ possesses a normal
series if there exist:

λ1m(x) ≤ λ2m(x) ≤ · · · ≤ λkm(x) = λm(x)

λ1n(x) ≥ λ2n(x) ≥ · · · ≥ λkn(x) = λn(x)

}
, (7)

∀x ∈ G such that λ1∗ = λ2∗ = · · · = λk∗ = λ∗ = G and λi ◁ λi+1 ∀ 1 ≤ i ≤ k. Eq. (7) can
be simply written as:

λ1 ⊆ λ2 ⊆ · · · ⊆ λk = λ (8)

such that λ1∗ = λ2∗ = · · · = λk∗ = λ∗ = G and λi ◁ λi+1 ∀ 1 ≤ i ≤ k or

λ1 ◁ λ2 ◁ · · · ◁ λk = λ (9)

for λ1∗ = λ2∗ = · · · = λk∗ = λ∗ = G.

Example 4. Using the IFSGs λ in Example 1, the normal series is:

λ1 ⊆ λ2 ⊆ λ3 ⊆ λ4 ⊆ λ5 ⊆ λ6 = λ,

where λi ◁ λi+1 ∀ 1 ≤ i ≤ k.



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 7 of 11

Example 5. Using the IFSGs of γ in Example 2, the normal series is:

γ1 ⊆ γ2 ⊆ γ3 ⊆ γ4 ⊆ γ5 = γ,

because γi ◁ γi+1 ∀ 1 ≤ i ≤ k.

Next, we define the concept of composition series for IFG.

Definition 15. Let G be a finite group and λ be an IFG of G. Then, λ possesses a
composition series if there exist:

λ1m(x) ≤ λ2m(x) ≤ · · · ≤ λkm(x) = λm(x)

λ1n(x) ≥ λ2n(x) ≥ · · · ≥ λkn(x) = λn(x)

}
, (10)

∀x ∈ G such that λ1∗ = λ2∗ = · · · = λk∗ = λ∗ = G with the properties

(i) λi ◁ λi+1 ∀ 1 ≤ i ≤ k,

(ii) λi+1/λi is simple ∀ 1 ≤ i ≤ k.

Eq. (10) can be simply written as:

λ1 ◁ λ2 ◁ · · · ◁ λk = λ (11)

for λ1∗ = λ2∗ = · · · = λk∗ = λ∗ = G and λi+1/λi is simple ∀ 1 ≤ i ≤ k.

Using the IFSGs of λ in Example 1, the composition series for λ is:

λ1 ◁ λ2 ◁ λ3 ◁ λ4 ◁ λ5 ◁ λ6 = λ,

where λi+1/λi is simple ∀ 1 ≤ i ≤ k.
Using the IFSGs of γ in Example 2, the composition series is:

γ1 ◁ γ2 ◁ γ3 ◁ γ4 ◁ γ5 = γ,

because λi+1/λi is simple ∀ 1 ≤ i ≤ k.

Theorem 2. Every finite IFG of a finite group has a composition series.

Proof. Let γ be an IFG of a finite group G. We present the proof using the principle
of induction. Assume every IFG of order less than |γ| has a composition series. If γ is
simple, then it has a trivial composition series. Otherwise, if γ is not simple, then it has
at least one non-trivial proper IFNSG and a maximal non-trivial IFNSG denoted as δ in
γ, since γ is finite. Then |δ| < |γ|.

By the principle of induction, |δ| has a composition series, which is:

δ1m(x) ≤ δ2m(x) ≤ · · · ≤ δkm(x) = δm(x)

δ1n(x) ≥ δ2n(x) ≥ · · · ≥ δkn(x) = δn(x)

}
, (12)



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 8 of 11

∀x ∈ G such that δ1∗ = δ2∗ = · · · = δk∗ = G with δi ◁ δi+1 ∀ 1 ≤ i ≤ k and δi+1/δi is
simple ∀ 1 ≤ i ≤ k.

But δ ◁ γ because δ is maximal in γ, then we have

δ1m(x) ≤ δ2m(x) ≤ · · · ≤ δkm(x) = δm(x) ≤ γm(x)

δ1n(x) ≥ δ2n(x) ≥ · · · ≥ δkn(x) = δn(x) ≥ γn(x)

}
(13)

∀x ∈ G, which proves that γ has a composition series.

Theorem 3 (The Jordan-Hölder Theorem). Every IFG of a finite group has at least two
equivalent composition series.

Proof. Let γ be an IFG of a finite group G. Assume γ has two composition series:

γ1m(x) ≤ γ2m(x) ≤ · · · ≤ γkm(x) = γm(x)

γ1n(x) ≥ γ2n(x) ≥ · · · ≥ γkn(x) = γn(x)

}
, (14)

δ1m(x) ≤ δ2m(x) ≤ · · · ≤ δlm(x) = δm(x)

δ1n(x) ≥ δ2n(x) ≥ · · · ≥ δln(x) = δn(x)

}
, (15)

such that γi ◁ γi+1 with γi+1/γi simple ∀ 1 ≤ i ≤ k and δi ◁ δi+1 with δi+1/δi simple ∀
1 ≤ i ≤ l. In each series, γ1∗ = γ2∗ = · · · = γk∗ = G and δ1∗ = δ2∗ = · · · = δl∗ = G.

Now, we show that k = l and (γ2/γ1, γ3/γ2, · · · , γk/γk−1) is identical to (δ2/δ1, δ3/δ2, · · · , δl/δl−1).
We establish the proof by induction on |γ|. The proof is trivial if |γ| = 1. Assume
γ(k−1) = δ(l−1), then the result is straightforward by induction. On the contrary, assume
γ(k−1) ̸= δ(l−1). Set A = γ(k−1), B = δ(l−1), and C = A∩B, where C is a maximal IFNSG
of γ(k−1) and δ(l−1).

But, C has a composition series,

C1m(x) ≤ C2m(x) ≤ · · · ≤ Ctm(x) = Cm(x)

C1n(x) ≥ C2n(x) ≥ · · · ≥ Ctn(x) = Cn(x)

}
, (16)

∀x ∈ G. Then,
γ1m(x) ≤ γ2m(x) ≤ · · · ≤ γ(k−1)m

(x) = Am(x)

γ1n(x) ≥ γ2n(x) ≥ · · · ≥ γ(k−1)n
(x) = An(x)

}
(17)

∀x ∈ G, and

C1m(x) ≤ C2m(x) ≤ · · · ≤ Ctm(x) = Cm(x) ≤ Am(x)

C1n(x) ≥ C2n(x) ≥ · · · ≥ Ctn(x) = Cn(x) ≤ An(x)

}
(18)

∀x ∈ G are the composition series for A. By induction, we have k− 1 = t+1 ⇒ k− 2 = t,
and

(γ2/γ1, γ3/γ2, · · · , γ(k−1)/γ(k−2)) ∼ (C2/C1, C3/C2, · · · , Ct/Ct−1, A/C). (19)



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 9 of 11

Likewise, we get

δ1m(x) ≤ δ2m(x) ≤ · · · ≤ δ(l−1)m
(x) = Bm(x)

δ1n(x) ≥ δ2n(x) ≥ · · · ≥ δ(l−1)n
(x) = Bn(x)

}
(20)

∀x ∈ G and
C1m(x) ≤ C2m(x) ≤ · · · ≤ Ctm(x) = Cm(x) ≤ Bm(x)

C1n(x) ≥ C2n(x) ≥ · · · ≥ Ctn(x) = Cn(x) ≤ Bn(x)

}
(21)

∀x ∈ G, which are the composition series for B. Thus, l − 1 = t+ 1 ⇒ l − 2 = t, and

(δ2/δ1, δ3/δ2, · · · , δ(l−1)/δ(l−2)) ∼ (C2/C1, C3/C2, · · · , Ct/C(t−1), B/C). (22)

From k− 1 = t+ 1 = l− 1, we have k = l. By attaching γ/A to both sides of (19), we
have

(γ2/γ1, · · · , γ(k−1)/γ(l−2), γ/γ(k−1)) ∼ (C2/C1, · · · , Ct/C(t−1), A/C, γ/A). (23)

Likewise, attaching γ/B to both sides of (22), we have

(δ2/δ1, · · · , δ(l−1)/δ(l−2), δ/δ(l−1)) ∼ (C2/C1, · · · , Ct/C(t−1), B/C, γ/B). (24)

The right hand side of (23) and (24) are equal except (A/C, γ/A) and (B/C, γ/B).
Hence, (A/C, γ/A) ∼ (B/C, γ/B) and so

(γ2/γ1, · · · , γk/γ(k−1)) ∼ (δ2/δ1, · · · , δl/δ(l−1)).

4. Conclusion

In this paper, the notions of simple IFGs, maximal normal IFSGs, normal series for
IFGs, and composition series for IFGs were defined and described with examples and
particular results. In addition, it was verified that every IFG of a finite group has a
composition series. Finally, the Jordan-Hölder Theorem was discussed in IFS context, and
it was proved that any two composition series of an IFG of a finite group are equivalent.
This explorations have further enrich the study of fuzzy algebra and pave the way for the
study of nilpotency under IFGs. In addition, this work can be extended to other variants
of FST to enhance algebra under uncertain environments.

Acknowledgements

We greatly appreciate the editor-in-chiefs and the reviewers for their valuable com-
ments, which have improved the quality of the work.



P. A. Ejegwa, N. Kausar, T. Cagin / Eur. J. Pure Appl. Math, 18 (2) (2025), 5908 10 of 11

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