EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5572 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Algebraic Structures of Complex Fermatean Fuzzy Subgroups Eman A. AbuHijleh1,∗, Abd Ulazeez Alkouri2 1 Department of Basic Sciences, Al-Zarqa University College, Al-Balqa Applied University, Jordan 2 Mathematics Department, Science College, Ajloun National University, Jordan Abstract. A complex Fermatean fuzzy set provides a detailed framework for representing a spe- cific type of information and has been effectively applied to decision-making problems. This study introduces complex Fermatean fuzzy subgroups (CFFSGs), an extension of Fermatean fuzzy sub- groups, and complex Pythagorean fuzzy subgroups. The key innovation of CFFSGs lies in their capacity to represent two variables within their algebraic structure, surpassing the capabilities of traditional Fermatean fuzzy subgroups. The research establishes the formal definition and prop- erties of CFFSGs, adapting them to a complex framework that incorporates amplitude and phase components. Additionally, the concepts of complex Fermatean fuzzy cosets and complex Fer- matean fuzzy normal subgroups are introduced. The study also investigates and examines the characteristics of homomorphisms between complex Fermatean fuzzy subgroups. 2020 Mathematics Subject Classifications: 08A35, 08A72, 20N25 Key Words and Phrases: Complex fermatean fuzzy subgroup, complex fermatean fuzzy normal subgroup, homomorphism of fermatean fuzzy subgroup 1. Introduction Zadeh [40] introduced the fuzzy set (FS) as a novel concept to address uncertainty and vagueness in decision-making (DM) problems. This foundational work sparked the publication of hundreds of studies that extended and applied FS concepts across various fields. A significant advancement was made by Atanassov in 1986 [12], who introduced the intuitionistic fuzzy set (IFS). Later, Yager [38] expanded this framework in 2013 by defining the Pythagorean fuzzy set (PFS), positioning FS as a subset of IFS, which in turn is a subset of PFS. In 2017, Yager further generalized these concepts by introducing q-rung orthopair fuzzy sets [39]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5572 Email addresses: emanhijleh@bau.edu.jo (E.A. AbuHijleh), alkouriabdulazeez@anu.edu.jo (A. Alkouri) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 2 of 19 This generalization was followed by specific cases, such as the Fermatean fuzzy set (FFS) defined by Senapati et al. in 2020 [33]. These sets differ in their constraints; for IFS, the sum of membership and non-membership degrees must lie between zero and one; for PFS, the sum of their squares must meet this condition; for FFS, it is the sum of their cubes; and for q-rung orthopair fuzzy sets, the sum of the Kth powers of membership and non-membership degrees must satisfy this constraint. Researchers have explored the properties and operations of these sets, developed new measures, and applied them to DM problems, as demonstrated in studies like [18], [23], and [27]. More recently, in 2022, M. Akram et al. [4] proposed a DM approach integrating the attributes of the traditional VIKOR method within the framework of a multidimensional complex Fermatean fuzzy N-soft set. Real-world problems often involve multiple variables, necessitating more advanced con- ceptual frameworks. This need led to the intellectual introduction of the complex fuzzy set (CFS) by Ramot et al. (2002) [30], which incorporates two key variables: amplitude and phase. Building on this concept, Alkouri and Salleh (2012) [8] introduced the com- plex intuitionistic fuzzy set (CIFS), enhancing the properties of CFS. Subsequent works have extended and refined these ideas, including contributions from Alkouri (2013) [9] and Ramot (2003) [29]. In 2019, Ullah et al. [36] developed the complex Pythagorean fuzzy set (CPFS) con- cept, introducing various associated measures. This was followed in 2020 by Liu et al. [25], who proposed complex q-Rung orthopair fuzzy sets, a generalization encompassing CIFS, CPFS, CFFS, and orthopair fuzzy sets. Further advancements were made in 2021 when Chinnadurai et al. [17] defined the complex Fermatean fuzzy set (CFFS) and explored its applications in decision-making (DM) problems. The application of complex fuzzy theories in DM has since expanded, as demonstrated by recent works such as Chen (2023) [16] and Wang (2023) [37]. Other notable general- izations include the use of complex neutrosophic graphs for hospital infrastructure design by Alqahtani et al. (2024) [10] and the application of complex hesitant fuzzy graphs by AbuHijleh et al. (2023) [2] and Alkouri (2023) [6]. Additionally, Alqaraleh et al. (2022) [11] introduced bipolar complex fuzzy soft sets with practical applications. For further exploration of generalizations and applications within this domain, refer to recent works by Al-Masarwah et al. (2023) [5], Fallat et al. (2022) [19], and Hazaymeh et al. (2024–2025) [21, 22]. In parallel with the advancements in fuzzy set theory, significant progress has also been made in fuzzy group theory. Rosenfeld (1971) [32] introduced the concept of a fuzzy subgroup (FSG) as a generalization of the classical group. This foundational work inspired numerous mathematicians to explore group theory through fuzzy sets. In 1989, Biswas [15] expanded on this idea by defining the intuitionistic fuzzy subgroup (IFSG) and analyzing its algebraic properties. In 2020, Bhunia et al. [14] introduced the Pythagorean fuzzy subgroup (PFSG), fur- ther examining its algebraic structure. The same year, manuscripts on the complex intu- itionistic fuzzy subgroup (CIFSG) [20] and the complex fuzzy subgroup (CFSG) [3] were published, adding new dimensions to the field. Most recently, a study on the complex E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 3 of 19 Pythagorean fuzzy subgroup (CPFSG) was published for publication [7]. Building on these developments, Silambarasan (2021) [35] introduced the Fermatean fuzzy subgroup (FFSG), analyzing its properties and its relationships with IFSG and PFSG. This work has inspired further research into FFSG, including studies by Balamu- rugan (2022) [13], Kalaichelvan (2022) [24], Nagarajan (2021) [26], Onasanya (2022) [28], and Muhammad (2022) [31]. Onasanya et al. (2022) [28] introduced Fermatean fuzzy subgroups within the q- rung orthopair fuzzy sets framework in group theory, coining the term ”harmonized fuzzy groups”. These groups unify various subgroup types as special cases and provide a versatile foundation for further exploration. While their study addressed several properties of harmonized fuzzy groups, the extension to complex harmonized fuzzy groups remains an open area of research. The motivation for constructing complex Fermatean fuzzy subgroups (CFFSG) lies in advancing the mathematical framework of fuzzy group theory. Specifically, it seeks to incorporate periodic information inherent in complex Fermatean fuzzy sets (CFFS) and leverage their capacity to represent larger values compared to complex Pythagorean fuzzy sets (CPFS) and complex intuitionistic fuzzy sets (CIFS). This development not only enriches the theoretical landscape but also opens pathways for practical applications, such as cryptographic primitives and generalized periodic algorithms. Future work includes the construction of CFFSG and the development of cyclic CFFSG as a specialized extension. Additionally, integrating results from fixed-point theory (e.g., [1] and [34]) with complex Fermatean fuzzy algebra presents a promising avenue for novel applications. Such integration could address real-world problems by employing metric space frameworks to create innovative solutions. This paper explores the concept of the complex Fermatean fuzzy subgroup (CFFSG) as an enhancement of both the complex Pythagorean fuzzy subgroup (CPFSG) and the Fermatean fuzzy subgroup (FFSG). Section 2 provides an overview of key definitions from relevant literature, establishing the foundational concepts. Section 3 introduces the for- mal definition of CFFSG and examines its fundamental properties. Section 4 extends the discussion to complex Fermatean fuzzy normal subgroups, detailing their characteris- tics. Section 5 analyzes homomorphisms within the context of CFFSG, highlighting their properties and implications. Finally, Section 6 summarizes the findings of this study and proposes potential directions for future research. 2. Preliminaries Zadeh defined fuzzy set in 1965 [40]. Definition 1. [40] A fuzzy set (FS) K of the universe of discourse X is defined by mem- bership function; K : X → [0, 1], whereas K(x) is a degree of membership for any x in X. Ramot et al. defined complex fuzzy set (CFS) on a crisp set in 2002 [30]. E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 4 of 19 Definition 2. [30] A complex fuzzy set (CFS) K of the universe of discourse X is defined by membership function; K(x) : X → {z : z ∈ C, |z| ≤ 1}, that assigns a degree of membership K(x) = p(x)e2πiω(x) for any x in X, where the value of K(x) is defined by two variables p(x) and ω(x) and both are located within zero and one. On the other hand, Atanassov (1986) [12] introduced the concept of an intuitionistic fuzzy set (IFS) by incorporating a non-membership degree, where the sum of the member- ship degree and the non-membership degree lies between zero and one. This framework was later expanded in various ways, one of which is the Pythagorean fuzzy set (PFS) [38], introduced by Yager in 2013, where the sum of the squares of the membership and non-membership degrees is constrained between zero and one. Subsequently, Senapti and Yager (2020) [33] defined the Fermatean fuzzy set (FFS), as described below. Definition 3. [33] Let X be a universe of discourse, then a fermatean fuzzy set F on X defined by F = {(x,K(x),L(x)) : x ∈ X}. Such that K(x) ∈ [0, 1] and L(x) ∈ [0, 1] are the degree of membership and the degree of non-membership for any x ∈ X, respectively, and 0 ≤ K3(x) + L3(x) ≤ 1, for all x ∈ X. Additionally, Yager (2017) [39] introduced a broader class of fuzzy sets known as q-rung orthopair fuzzy sets, in which the sum of the qth powers of the membership and non- membership degrees is constrained between zero and one. He demonstrated that as the value of q increases, the space of acceptable orthopair sets expands, thereby providing users with greater flexibility in expressing their beliefs regarding the degree of member- ship. On the other hand, the concept of the complex intuitionistic fuzzy set (CIFS) was introduced in 2012 [8], followed by the complex Pythagorean fuzzy set (CPFS) in 2019 [36], and more recently, the complex Fermatean fuzzy set (CFFS) was presented in 2021 [17]. These advancements involve extending traditional fuzzy sets to complex fuzzy sets for both membership and non-membership degrees. The definition of the complex Fermatean fuzzy set (CFFS) is provided below. Definition 4. [17] Let X be a universe of discourse and defined a complex fermatean fuzzy set ϕ on X, where ϕ = {(x,K(x),L(x)) : x ∈ X}. Such that K(x) : X → {z : z ∈ C, |z| ≤ 1} and L(x) : X → {z : z ∈ C, |z| ≤ 1}, are the degree of membership and non- membership of x ∈ X, respectively. Moreover, K(x) = p(x)e2πiω(x) and L(x) = q(x)e2πiν(x) are satisfying the conditions; 0 ≤ p3(x) + q3(x) ≤ 1 and 0 ≤ ω3(x) + ν3(x) ≤ 1. In the previous definition, if conditions become 0 ≤ pk(x) + qk(x) ≤ 1 and 0 ≤ ωk(x) + νk(x) ≤ 1, for all x ∈ X with k ≥ 1. Then ϕ define a complex q-rung orthopair fuzzy sets (2020) [25]. Another approach to fuzzy sets is the concept of fuzzy subgroups, first introduced by Rosenfeld in 1971 [32]. This concept was subsequently enhanced with the introduction of E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 5 of 19 the intuitionistic fuzzy subgroup (IFSG) in 1989 [15]. In 2021, E.A. Abuhijleh et al. [3] defined the complex fuzzy subgroup (CFSG), while the complex intuitionistic fuzzy sub- group (CIFSG) was introduced in 2020 [20]. The Pythagorean fuzzy subgroup (PFSG) was presented in 2020 [14], followed by the complex Pythagorean fuzzy subgroup (CPFSG) in 2023 [7]. Additionally, the Fermatean fuzzy subgroup (FFSG) was defined in 2021 [35], with Onasanya et al. (2022) [28] also contributing to the development of the Fermatean fuzzy subgroup. Here, we produce definitions of FSG, CFSG, and FFSG, respectively. Definition 5. [32] Let K : X → [0, 1] defined a fuzzy subset of a group (X, ∗). Then K presented a fuzzy subgroup (FSG) of (X, ∗), if the following conditions hold: i) K(x ∗ y) ≥ K(x) ∧K(y). ii) K(x−1) ≥ K(x), for all x, y ∈ X Definition 6. [3] Let K(x) : X → {z : z ∈ C, |z| ≤ 1} be a complex fuzzy subset of a group (X, ∗). Then K presented a complex fuzzy subgroup, of (X, ∗), if the following conditions hold: i) K(x ∗ y) ≥ K(x) ∧K(y). ii) K(x−1) ≥ K(x), for all x, y ∈ X Equivalently, for any x, y ∈ X and K(x) = p(x)e2πiω(x), we have: i) p(x ∗ y) ≥ p(x) ∧ p(y) and ω(x ∗ y) ≥ ω(x) ∧ ω(y). ii) p(x−1) ≥ p(x) and ω(x−1) ≥ ω(x). Definition 7. [35] Let (X, ∗) be a group and F = (K,L) be a fermatean fuzzy set of X. Then F is a fermatean fuzzy subgroup of X if the following conditions hold: (i) K3(x ∗ y) ≥ K3(x) ∧K3(y) and L3(x ∗ y) ≤ L3(x) ∨ L3(y). (ii) K3(x−1) ≥ K3(x) and L3(x−1) ≤ L3(x) , ∀ x, y ∈ X In the previous definition, if the power k = 1, 2 we get intuitionistic fuzzy subgroup (IFSG) [15] and Pythagorean fuzzy subgroup (PFSG) [14], respectively. The union, intersection, and complement of CFFS defined in 2021 [17], as follows. Definition 8. Let ϕ1 = (K1,L1) and ϕ2 = (K2,L2) be two CFFSs on X, where: Kj(x) : X → {pj(x)e2πiωj(x) : 0 ≤ pj(x), ωj(x) ≤ 1}, and Lj(x) : X → {qj(x)e2πiνj(x) : 0 ≤ qj(x), νj(x) ≤ 1}, for j = 1, 2, then: 1. ϕ1 ∩ ϕ2 = (K1 ∩K2,L1 ∩ L2), where (K1 ∩K2)(x) = (p1(x) ∧ p2(x))e 2πi(ω1(x)∧ω2(x)) and (L1 ∩ L2)(x) = (q1(x) ∨ q2(x))e 2πi(ν1(x)∨ν2(x)). E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 6 of 19 2. ϕ1 ∪ ϕ2 = (K1 ∪K2,L1 ∪ L2), where (K1 ∪K2)(x) = (p1(x) ∨ p2(x))e 2πi(ω1(x)∨ω2(x)) and (L1 ∪ L2)(x) = (q1(x) ∧ q2(x))e 2πi(ν1(x)∧ν2(x)). 3. ϕc = ϕ̄ = (L,K) = (q(x)e2πiν(x), p(x)e2πiω(x)). 3. Complex Fermatean Fuzzy Subgroups A generalization of Fermatean fuzzy subgroups [35] and complex Fermatean fuzzy sets [17] is presented in the following definition, which also serves as a generalization of complex Pythagorean fuzzy subgroups [7]. Definition 9. Let (X, ∗) be a group and ϕ = (p e2πiω, q e2πiν) be a CFFS of X. Then ϕ is complex fermatean fuzzy subgroup (CFFSG) of X, where p3 + q3 ≤ 1 and ω3 + ν3 ≤ 1, if the following holds: 1a. p3(x ∗ y)e2πiω 3(x ∗ y) ≥ p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y). where, p3(x ∗ y) ≥ p3(x) ∧ p3(y) and ω3(x ∗ y) ≥ ω3(x) ∧ ω3(y). 1b. q3(x ∗ y)e2πiν 3(x ∗ y) ≤ q3(x)e2πiν 3(x) ∨ q3(y)e2πiν 3(y) where, q3(x ∗ y) ≤ q3(x) ∨ q3(y) and ν3(x ∗ y) ≤ ν3(x) ∨ ν3(y) 2a. p3(x−1)e2πiω 3(x−1) ≥ p3(x)e2πiω 3(x) where, p3(x−1) ≥ p3(x) and ω3(x−1) ≥ ω3(x). 2b. q3(x−1)e2πiν 3(x−1) ≤ q3(x)e2πiν 3(x) where, q3(x−1) ≤ q3(x) and ν3(x−1) ≤ ν3(x) In the previous definition, if the power k = 1, 2 we get a complex intuitionistic fuzzy subgroups (CIFSGs) [20], and a complex Pythagorean fuzzy subgroups (CPFSGs) [7], respectively. Proposition 1. Let ϕ = (p e2πiω, q e2πiν) be a CFFSG of a group (X, ∗), then the following holds: (i) p3(id)e2πiω 3(id) ≥ p3(x)e2πiω 3(x), where p3(id) ≥ p3(x) and ω3(id) ≥ ω3(x). (ii) q3(id)e2πiν 3(id) ≤ q3(x)e2πiν 3(x), where q3(id) ≤ q3(x) and ν3(id) ≤ ν3(x). (iii) p3(x−1)e2πiω 3(x−1) = p3(x)e2πiω 3(x), where p3(x−1) = p3(x) and ω3(x−1) = ω3(x). (iv) q3(x−1)e2πiν 3(x−1) = q3(x)e2πiν 3(x), where q3(x−1) = q3(x) and ν3(x−1) = ν3(x). for all x ∈ X, where id is the identity of all elements. Proof. Since ϕ is CFFSG then by Definition 9: ”1” and ”2” can be proved as follow, p3(id)e2πiω 3(id) = p3(x o x−1)e2πiω 3(x o x−1) ≥ min{p3(x)e2πiω3(x) , p3(x−1)e2πiω 3(x−1)}= min{p3(x), p3(x−1)}e2πimin{ω3(x),ω3(x−1)} = p3(x)e2πiω 3(x). E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 7 of 19 In addition, q3(id)e2πiν 3(id) = q3(x o x−1)e2πiν 3(x o x−1) ≤ max{q3(x)e2πiω3 q (x), q3(x−1)e2πiω 3 q (x −1)} = max{q3(x), q3(x−1)}e2πimax{ν3(x),ν3(x−1)} = q3(x)e2πiν 3(x). ”3” and ”4” can be prove in the same manner of ”1” and ”2”. The following is an example of complex fermatean fuzzy subgroup (CFFSG). Example 1. For the set D2 which define a dihedral group of order four and isomorphic to the direct sum of two cyclic group Z2. Hence, D2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. Also, let ϕ = (K,L) be a CFFS on D2, such that: ϕ((0, 0)) = (0.8e2πi(0.85), 0.5e2πi(0.6)), ϕ((1, 0)) = (0.78e2πi(0.83), 0.6e2πi(0.65)), ϕ((0, 1)) = (0.75e2πi(0.8), 0.65e2πi(0.68)) = ϕ(xy). At first, by Definition 4, it is easy to check that ϕ is CFFS, but it is not CPFS. For exam- ple in ϕ((0, 0)), we have ω3+ν3 = (0.614+0.216) = 0.83 ≤ 1 and ω2+ν2 = 0.723+0.36 ≰ 1. In the second part, it suffices to prove that the set ϕ(ℓ) = (p(ℓ)e2πiω(ℓ), q(ℓ)e2πiν(ℓ)) is CFFSG on D2, for any ℓ ∈ D2. So that check conditions (1a) and (1b) in the Definition 9, as follows: Consider ℓ1 = (1, 0), ℓ2 = (0, 1), then (1, 0) ⊕ (0, 1) = (1, 1) 1a. p3((1, 0) ⊕ (0, 1)) = p3((1, 1)) = 0.422 ≥ min{p3((1, 0)), p3((0, 1))} = 0.422 and ω3((1, 0)⊕(0, 1)) = ω3((1, 1)) = 0.512 ≥ min{ω3((1, 0)), ω3((0, 1))} = 0.512. 1b. q3((1, 0) ⊕ (0, 1)) = q3((1, 1)) = 0.275 ≤ max{q3((1, 0)), q3((0, 1))} = 0.275 and ν3((1, 0)⊕ (0, 1)) = ν3((1, 1)) = 0.314 ≤ max{ν3((1, 0)), ν3((0, 1))} = 0.314. Then the property satisfied at ℓ1 = (1, 0), ℓ2 = (0, 1), and one can go through all ℓi and check conditions. In addition, for conditions (2a) and (2b) in the Definition 9, they satisfied too, since ℓ = ℓ−1 for any ℓ ∈ D2. Note that, it is easy to see that ϕ is satisfied the previous proposition. The following theorem prove that any CPFSG is a CFFSG. Theorem 1. If ϕ is a CPFSG of the group (X, ∗), then ϕ is a CFFSG of the group (X, ∗). Proof. At first, to show that p3(x ∗ y)e2πiω 3(x ∗ y) ≥ p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y) and q3(x ∗ y)e2πiν 3(x ∗ y) ≤ q3(x)e2πiν 3(x) ∨ q3(y)e2πiν 3(y). We know that ϕ is a CPFSG, then p2(x ∗ y)e2πiω 2(x ∗ y) ≥ p2(x)e2πiω 2(x) ∧ p2(y)e2πiω 2(y) and q2(x ∗ y)e2πiν 2(x ∗ y) ≤ q2(x)e2πiν 2(x) ∨ q2(y)e2πiν 2(y), where p2 + q2 ≤ 1 and ω2 + ν2 ≤ 1. Then, we have four cases to consider: a) Let p2(x)e2πiω 2(x) ≥ p2(y)e2πiω 2(y) and q2(x)e2πiν 2(x) ≥ q2(y)e2πiν 2(y), then p2(x ∗ y)e2πiω 2(x ∗ y) ≥ p2(y)e2πiω 2(y). Now consider p3(x ∗ y)e2πiω 3(x ∗ y) ≥ p3(y)e2πiω 3(y) = p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y). Moreover, q2(x ∗ y)e2πiν 2(x ∗ y) ≤ q2(x)e2πiν 2(x). Now consider q3(x ∗ y)e2πiν 3(x ∗ y) ≤ q3(x)e2πiν 3(x) = q3(x)e2πiν 3(x) ∨q3(y)e2πiωq 3(y). E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 8 of 19 b) Let p2(x)e2πiω 2(x) ≤ p2(y)e2πiω 2(y) and q2(x)e2πiν 2(x) ≤ q2(y)e2πiν 2(y), then with same argument of case a, we get the result. c) Let p2(x)e2πiω 2(x) ≤ p2(y)e2πiω 2(y) and q2(x)e2πiν 2(x) ≥ q2(y)e2πiν 2(y), then p2(x ∗ y)e2πiω 2(x ∗ y) ≥ p2(x)e2πiω 2(x). Now consider p3(x ∗ y)e2πiω 3(x ∗ y) ≥ p3(x)e2πiω 3(x) = p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y). Moreover, q2(x ∗ y)e2πiν 2(x ∗ y) ≤ q2(x)e2πiν 2(x). Now consider q3(x ∗ y)e2πiν 3(x ∗ y) ≤ q3(x)e2πiν 3(x) = q3(x)e2πiν 3(x) ∨q3(y)e2πiν3(y). d) Let p2(x)e2πiω 2(x) ≥ p2(y)e2πiω 2(y) and q2(x)e2πiν 2(x) ≤ q2(y)e2πiν 2(y), then with same argument of case c, we get the result. Secondly, since p2(x−1)e2πiω 2(x−1) ≥ p2(x)e2πiω 2(x) and q2(x−1)e2πiν 2(x−1) ≤ q2(x)e2πiν 2(x), then p3(x−1)e2πiω 3(x−1) ≥ p3(x)e2πiω 3(x) and q3(x−1)e2πiν 3(x−1) ≤ q3(x)e2πiν 3(x) too. The converse of Theorem 1 is not always true, please see the following example. Example 2. For the set X = {1,−1, i,−i}, define a group (X, .), where ′.′ is the known multiplication. Also define ϕ = (K,L) be a CFFS on X, where: ϕ(1) = (0.8e2πi(0.75), 0.3e2πi(0.3)), ϕ(−1) = (0.8e2πi(0.5), 0.65e2πi(0.45)), ϕ(i) = (0.7e2πi(0.45), 0.9e2πi(0.6) = ϕ(−i). Now, it is easy to check that for all x ∈ X, that ϕ is CFFS, c.f. Definition 4. But, ϕ is not CPFS, for example at x = −1 we have p2 + q2 = 0.64 + 0.4225 ≰ 1. Now, it suffices to prove that the set ϕ(x) = (p(x) e2πiω(x), q(x) e2πiν(x)) is CFFSG: i) First, for any x, y ∈ X, we check that: a. p3(x ∗ y) ≥ min{p3(x), p3(y)} and ω3(x ∗ y) ≥ min{ω3(x), ω3(y)}. b. q3(x ∗ y) ≤ max{q3(x), q3(y)} and ν3(x ∗ y) ≤ max{ν3(x), ν3(y)}. Hence, consider x = i, y = −i, then i.− i = 1: a. p3(i ∗ −i) = p3(1) = 0.512 ≥ min{p3(i), p3(−i)} = 0.343 and ω3(i ∗ −i) = ω3(1) = 0.42188 ≥ min{ω3(i), ω3(−i)} = 0.09112. b. q3(i ∗ −i) = q3(1) = 0.027 ≤ max{q3(i), q3(−i)} = 0.729 and ν3(i ∗ −i) = ν3(1) = 0.027 ≤ max{ν3(i), ν3(−i)} = 0.216. , then the property satisfied at x = i, y = −i; one can go through all x and see that property satisfied. ii) Second, since 1 = 1−1, −1 = −1−1 and i = −i−1, where ϕ(i) = ϕ(−i), then property 2 in of CFFSG is satisfied too. Note that, since CIFSG is subclass of CPFSG [7], then CIFSG is subclass of CFFSG. Proposition 2. For a CFFS ϕ = (p e2πiω, q e2πiν) of a group (X, ∗), it is a CFFSG if and only if: 1. p3(x ∗ y−1)e2πiω 3(x ∗ y−1) ≥ p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y) , where p3(x ∗ y−1) ≥ p3(x) ∧ p3(y) and ω3(x ∗ y−1) ≥ ω3(x) ∧ ω3(y) E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 9 of 19 2. q3(x ∗ y−1)e2πiν 3(x ∗ y−1) ≤ q3(x)e2πiν 3(x) ∨ q3(y)e2πiν 3(y) , where q3(x ∗ y−1) ≤ q3(x) ∨ q3(y) and ν3(x ∗ y−1) ≤ ν3(x) ∨ ν3(y) Proof. (=⇒) According to Proposition 1, we have p3(x−1)e2πiω 3(x−1) = p3(x)e2πiω 3(x) and q3(x−1)e2πiν 3(x−1) = q3(x)e2πiν 3(x) for all x ∈ X, then results follow by Definition 9. (⇐=) First, ϕ is CFFS and is defined on group (X, ∗), then: (i) p3(id)e2πiω 3(id) = p3(x ∗ x−1)e2πiω 3(x ∗ x−1) ≥ p3(x)e2πiω 3(x), where p3(x ∗ x−1) ≥ p3(x) and ω3(x ∗ x−1) ≥ ω3(x). (ii) p3(x−1)e2πiω 3(x−1) = p3(id ∗ x−1)e2πiω 3(id ∗ x−1) ≥ min{p3(id)e2πiω3(di), p3(x)e2πiω 3(x)} = min{p3(id), p3(x)}e2πimin{ω3(di),ω3(x)} = p3(x)e2πiω 3(x), by (i). (ii) p3(x ∗ y)e2πiω 3(x ∗ y) = p3(x ∗ (y−1)−1)e2πiω 3(x ∗ (y−1)−1) ≥ p3(x)e2πiω 3(x) ∧ p3(y−1)e2πiω 3(y−1) ≥ p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y), by (ii). Similarly, we have: (iv) q3(id)e2πiν 3(id) = q3(x ∗ x−1)e2πiν 3(x ∗ x−1) ≤ q3(x)e2πiν 3(x), where q3(x ∗ x−1) ≤ q3(x) and ν3(x ∗ x−1) ≤ ν3(x). (v) q3(x−1)e2πiν 3(x−1) = q3(id ∗ x−1)e2πiν 3(id ∗ x−1) ≤ max{q3(id)e2πiν3(di), q3(x)e2πiν3(x)} = max{q3(id), q3(x)}e2πimax{ν3(di),ν3(x)} = q3(x)e2πiν 3(x), by (iv). (vi) q3(x ∗ y)e2πiν 3(x ∗ y) = q3(x ∗ (y−1)−1)e2πiν 3(x ∗ (y−1)−1) ≤ q3(x)e2πiν 3(x) ∨ q3(y−1)e2πiν 3(y−1) ≤ q3(x)e2πiν 3(x) ∨ q3(y)e2πiν 3(y), by (v). Finally, by (iii) and (vi) the first condition was satisfied, and by (ii) and (v) the second condition was satisfied in the Definition 9, hence ϕ is CFFSG of a group (X, ∗). Proposition 3. The intersection of two CFFSGs of a group (X, ∗) is a CFFSG. Proof. Let S1, S2 be two CFFSGs of X and using previous proposition, then: i) p3S1∩S2 (x ∗ y−1)e 2πiω3 S1∩S2 (x ∗ y−1) = (p3S1 (x ∗ y−1) ∧ p3S2 (x ∗ y−1))e 2πi(ω3 S1 (x ∗ y−1)∧ ω3 S2 (x ∗ y−1)) ≥ (min{p3S1 (x), p3S1 (y)} ∧ min{p3S2 (x), p3S2 (y)})e2πi(min{ω3 S1 (x),ω3 S1 (y)}∧ min{ω3 S2 (x),ω3 S2 (y)}) = (min{p3S1 (x), p3S2 (x)} ∧ min{p3S1 (y), p3S2 (y)})e2πi(min{ω3 S1 (x),ω3 S2 (x)}∧ min{ω3 S1 (y),ω3 S2 (y)}) = (p3S1∩S2 (x) ∧ p3S1∩S2 (y))e 2πi(ω3 S1∩S2 (x)∧ ω3 S1∩S2 (y)) = p3S1∩S2 (x)e 2πiω3 S1∩S2 (x) ∧ p3S1∩S2 (y)e 2πiω3 S1∩S2 (y) . ii) q3S1∩S2 (x ∗ y−1)e 2πiν3S1∩S2 (x ∗ y−1) = (q3S1 (x ∗ y−1) ∨ q3S2 (x ∗ y−1))e 2πi(ν3S1 (x ∗ y−1)∨ ν3S2 (x ∗ y−1)) ≤ (max{q3S1 (x), q3S1 (y)} ∨ max{q3S2 (x), q3S2 (y)})e2πi(max{ν3S1 (x),ν3S1 (y)}∨ max{ν3S2 (x),ν3S2 (y)}) = (max{q3S1 (x), q3S2 (x)} ∨ max{q3S1 (y), q3S2 (y)})e2πi(max{ν3S1 (x),ν3S2 (x)}∨ max{ν3S1 (y),ν3S2 (y)}) = (q3S1∩S2 (x) ∨ q3S1∩S2 (y))e 2πi(ν3S1∩S2 (x)∨ν3S1∩S2 (y)) E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 10 of 19 = q2A∩B(x)e 2πiν2A∩B(x) ∨ q2A∩B(y)e 2πiν2A∩B(y). The union of two CFFSG is not necessary a CFFSG, see the following example. Example 3. Let (X, ∗) = (Z,+) be a group, also ϕ1 = 5Z and ϕ2 = 2Z be two CFFSG of Z. Where, ϕj = (Kϕj = pϕj (x)e 2πiωϕj (x) , Lϕj = qϕj (x)e 2πiνϕj (x)); j = 1, 2. They defined by: ϕ1(x) = (Kϕ1 ,Lϕ1) = { (0.8e2πi 0.6 , 0.7e2πi 0.3) : x ∈ 5Z (0.0e2πi 0.0 , 0.5e2πi 0.4) : elsewhere ϕ2(x) = (Kϕ2 ,Lϕ2) = { (0.8e2πi 0.7 , 0.4e2πi 0.2) : x ∈ 7Z (0.1e2πi 0.5 , 0.6e2πi 0.9) : elsewhere Then, we get: ϕ = ϕ1 ∪ ϕ2 =  (0.8e2πi 0.7 , 0.4e2πi 0.2) : x ∈ 7Z (0.8e2πi 0.6 , 0.7e2πi 0.3) : x ∈ 5Z− 7Z (0.0e2πi 0.0 , 0.5e2πi 0.4) : elsewhere Now, it is easy to check that ϕ is CFFS. Then, consider definition 9 to check if ϕ is CFFSG: For x1 = 15 and x2 = −7, then: < K3 ϕ(15 + −7), L3 ϕ(15 + −7) > = < K3 ϕ(8), L3 ϕ(8) > = < 0.0e2πi 0.0, 0.125e2πi 0.064 > and, < K3 ϕ(15) ∧K3 ϕ(−7), L3 ϕ(15) ∨ L3 ϕ(−7) > = < 0.512e2πi 0.216 ∧ 0.512e2πi 0.343, 0.343e2πi 0.027 ∨ 0.064e2πi 0.008 > = < 0.512e2πi 0.216, 0.343e2πi 0.027 > But, K3 ϕ(15 + −7) ≱ K3 ϕ(15) ∧K3 ϕ(−7); 0.0e2πi 0.0 ≱ 0.512e2πi 0.216 and L3 ϕ(15 + −7) ≰ L3 ϕ(15) ∨ L3 ϕ(−7); 0.125e2πi 0.064 ≰ 0.343e2πi 0.027. Therefore, ϕ = φ1 ∪ φ2 is not a CFFSG of (Z,+). Proposition 4. For a CFFS ϕ = (pe2πiω, qe2πiν) of a group (X, ∗). Then p3(x ∗ x ∗ · · · ∗ x)e2πiω 3(x∗x∗···∗x) ≥ p3(x)e2πiω 3(x), where p3(x ∗x ∗ · · · ∗x) ≥ p3(x) and ω3(x ∗x ∗ · · · ∗x) ≥ ω3(x). Also, q3(x∗x∗· · ·∗x)e2πiν3(x∗x∗···∗x) ≤ q3(x)e2πiν 3(x), where q3(x∗x∗· · ·∗x) ≤ q3(x) and ν3(x ∗ x ∗ · · · ∗ x) ≤ ν3(x). Proof. By induction the results will follow, such that p3(x∗x)e2πiω3(x∗x) ≥ p3(x)e2πiω 3(x), where p3(x ∗ x) ≥ p3(x) and ω3(x ∗ x) ≥ ω3(x). Also, q3(x ∗ x)e2πiν3(x∗x) ≤ q3(x)e2πiν 3(x), where q3(x ∗ x) ≤ q3(x) and ν3(x ∗ x) ≤ ν3(x). Theorem 2. For a CFFS ϕ = (pe2πiω, qe2πiν) of a group (X, ∗). The set M = {x ∈ X : p3(id)e2πiω 3(id) = p3(x)e2πiω 3(x) and q3(id)e2πiν 3(id) = q3(x)e2πiν 3(x)}, is a subgroup of X, where id is the identity of it. E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 11 of 19 Proof. At first, we have id ∈ M, hence M is not empty. Moreover, we need to show that x ∗ y−1 ∈ M for all x, y ∈ X. Assume that x, y ∈ M, where ϕ is CFFSG of X, then, by Proposition 2, p3(x∗y−1)e2πiω 3(x∗y−1) ≥ p3(x)e2πiω 3(x) ∧ p3(y)e2πiω 3(y) = p3(id)e2πiω 3(id), according to definition of M. But, x ∗ y−1 ∈ ϕ, hence p3(id)e2πiω 3(id) ≥ p3(x ∗ y−1)e2πiω 3(x∗y−1)), by Proposition 1. So that equality holds and p3(id)e2πiω 3(id) = p3(x∗y−1)e2πiω 3(x∗y−1)). Similarly, we can prove that q3(id)e2πiν 3(id) = q3(x ∗ y−1)e2πiν 3(x∗y−1), by Proposition 1 and Proposition 2. So that x ∗ y−1 ∈ M and M is subgroup of X. 4. Complex Fermatean fuzzy normal subgroup In this section, we define complex fermatean fuzzy normal subgroup (CFFNSG) and give equivalent conditions and some properties for it. Definition 10. Let ϕ = (pe2πiω, qe2πiν) be a CFFSG of a group (X, ∗). Then for z ∈ X, the complex fermatean fuzzy left coset of ϕ is the CFFS zϕ = ((zp)e2πi(zω), (zq)e2πi(zν)), which defined for membership by, (zp3(x))e2πi(zω 3(x)) = p3(z−1 ∗ x)e2πiω 3(z−1 ∗x). Also, for nonmembership it is defined by, (zq3(x))e2πi(zν 3(x)) = q3(z−1 ∗ x)e2πiν3(z−1 ∗x). In the same manner, the complex fermatean fuzzy right coset of ϕ is the CFFS ϕz = ((pz)e2πi(ωz), (qz)e2πi(νz)) and is defined by (p3(x)z)e2πi(ω 3(x)z) = p3(x ∗ z−1)e2πiω 3(x ∗ z−1) and (q3(x)z)e2πi(ν 3(x)z) = q3(x ∗ z−1)e2πiν 3(x ∗ z−1), for membership and nonmembership, respectively. Definition 11. Let ϕ = (p(x)e2πiω(x), q(x)e2πiν(x)) be a CFFSG of a group (X, ∗). Then ϕ is a complex fermatean fuzzy normal subgroup, of the group (X, ∗) if every complex fermatean fuzzy left coset is complex fermatean fuzzy right coset of ϕ in X, equivalently, zϕ = ϕz. Example 4. Let (X, ∗) = (Z3,+3) be a group with addition integer modulo 3. Define a CFFS ϕ, as follows: ϕ = (K(x), L(x)) =  (0.9e2πi 0.7, 0.8e2πi 0.8) : x = 0 (0.8e2πi 0.8, 0.7e2πi 0.6) : x = 1 (0.3e2πi 0.6, 0.5e2πi 0.6) : x = 2 First, it is easy to see at x = 0, ϕ is CFFS and not CPFS (so, not CIFS too). Second, to prove that ϕ is CFFNSG: Assume that z = 2 and x = 0 and take 2ϕ, then < (2K)3(0), (2L)3(0) > =< K3(2−1+30), L3(2−1+30) > =< p3(2−1+30)e 2πiω3(2−1+30), q3(2−1+30)e 2πiν3(2−1+30) > =< p3(1+30)e 2πiω3(1+30), q3(1+30)e 2πiν3(1+30) > =< p3(1)e2πiω 3(1), q3(1)e2πiν 3(1) > E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 12 of 19 =< p3(0+31)e 2πiω3(0+31), q3(0+31)e 2πiν3(0+31) > =< p3(0+32 −1)e2πiω 3(0+32 −1), q3(0+32 −1)e2πiν 3(0+32 −1) > =< K3(0+32 −1),L3(0+32 −1) > =< (K2)3(0), (L2)3(0) > So that, 2ϕ = ϕ2. Hence, for all x = 0, 1, 2 we can verify that zϕ = ϕz, where z ∈ X, i.e. ϕ is CFFNSG of the group (Z3,+3). Proposition 5. Let ϕ = (pe2πiω, qe2πiν) be a CFFSG of a group (X, ∗). Then ϕ is a CFFNSG of X if and only if p3(z1 ∗ z2)e 2πiω3(z1 ∗ z2) = p3(z2 ∗ z1)e 2πiω3(z2 ∗ z1) and q3(z1 ∗ z2)e 2πiν3(z1 ∗ z2) = q3(z2 ∗ z1)e 2πiν3(z2 ∗ z1). Proof. ⇒Assume that ϕ is a CFFNSG of X, then (z2p 3(z1))e 2πi(z2ω3(z1)) = (p3(z1)z2)e 2πi(ω3(z1)z2), for all z1, z2 ∈ X. Equivalently, p3(z−1 2 ∗ z1)e 2πiω3(z−1 2 ∗ z1) = p3(z1 ∗ z−1 2 )e2πiω 3(z1 ∗ z−1 2 ). Hence, p3(z2 ∗ z1)e2πiω 3(z2 ∗ z1) = p3((z−1 2 )−1 ∗ z1)e2πiω 3((z−1 2 )−1 ∗ z1) = p3(z1 ∗ (z−1 2 )−1)e2πiω 3(z1 ∗ (z−1 2 )−1) = p3(z1 ∗ z2)e 2πiω3(z1 ∗ z2). Similarly, we can verify that q3(z1 ∗ z2)e 2πiν3(z1 ∗ z2) = q3(z2 ∗ z1)e 2πiν3(z2 ∗ z1). ⇐Assume that z3 = z−1 1 , then for arbitrary z1, z2 ∈ X. We have p3(z1 ∗ z2)e2πiω 3(z1 ∗ z2) = p3(z2 ∗ z1)e 2πiω3(z2 ∗ z1), hence p3(z−1 3 ∗ z2)e 2πiω3(z−1 3 ∗ z2) = p3(z2 ∗ z−1 3 )e2πiω 3(z2 ∗ z−1 3 ) for any z3, z2 ∈ X. So that, (z3p 3(z2))e 2πi(z3ω3(z2)) = (p3(z2)z3)e 2πi(ω3(z2)z3). Similarly, we can prove that (z3q 3(z2))e 2πi(z3ν3(z2)) = (q3(z2)z3)e 2πi(ν3(z2)z3), then z3ϕ = ϕz3 for any z3 ∈ X, which implies that ϕ is CFFNSG of a group (X, ∗). Proposition 6. For a group (X, ∗) that was defined on CFFSG, ϕ = (pe2πiω, qe2πiν). Then ϕ is a CFFNSG of X if and only if p3(x)e2πiω 3(x) = p3(z ∗ x ∗ z−1)e2πiω 3(z∗x∗z−1), and q3(x)e2πiν 3(x) = q3(z ∗ x ∗ z−1)e2πiν 3(z∗x∗z−1), for all z, x ∈ X Proof. First consider, p3(x)e2πiω 3(x) = p3(x∗id)e2πiω3(x∗id) = p3(x∗z∗z−1)e2πiω 3(x∗z∗z−1) = p3(x ∗ (z ∗ z−1))e2πiω 3(x∗(z∗z−1)) = p3((x ∗ z) ∗ z−1)e2πiω 3((x∗z)∗z−1) = p3(z−1 ∗ (x ∗ z))e2πiω 3(z−1∗(x∗z)), whereas ϕ is CFFNSG of X. But z = (z−1)−1 and by similarity p3(x)e2πiω 3(x) = p3(z ∗x∗ z−1)e2πiω 3(z∗x∗z−1). Also, it is easy to show that q3(x)e2πiν 3(x) = q3(z ∗ x ∗ z−1)e2πiν 3(z∗x∗z−1) too. Conversely, p3(z∗x)e2πiω3(z∗x) = p3(z∗x∗id)e2πiω3(z∗x∗id) = p3(z∗(x∗z)∗z−1) e2πiω 3(z∗(x∗z)∗z−1) = p3(x∗z)e2πiω3(x∗z). Also, it is easy to show that q3(z ∗x)e2πiν3(z∗x) = q3(x∗z)e2πiν3(x∗z). Then by previous proposition, ϕ is CFFNSG of X. Theorem 3. Let ϕ be a CFFNSG of a group (X, ∗). Then the set M = {y ∈ X : p3(id)e2πiω 3(id) = p3(y)e2πiω 3(y) and q3(id)e2πiν 3(id) = q3(y)e2πiν 3(y)}, is a normal sub- group of X, where id is the identity of it. E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 13 of 19 Proof. At first id ∈ M, i.e. M is not empty. Moreover, it is subgroup of X, by Theorem 2. So that, p3(id)e2πiω 3(id) = p3(y ∗ z−1)e2πiω 3(y ∗ z−1) and q3(id)e2πiν 3(id) = q3(y ∗ z−1)e2πiν 3(y ∗ z−1). But, ϕ is a CFFNSG of (X, ∗). Then p3(y ∗ z−1)e2πiω 3(y ∗ z−1) = p3(z−1 ∗ y)e2πiω 3(z−1 ∗ y) and q3(y ∗ z−1)e2πiν 3(y ∗ z−1) = q3(z−1 ∗ y)e2πiν 3(z−1 ∗ y). Hence, (z−1 ∗ y) ∈ M and M is a normal subgroup of M. 5. Homomorphism on complex Fermatean fuzzy subgroup In this section, we discuss the effect of homomorphism on CFFSG. Definition 12. A homomorphism function h : X → U from group X to group U. Let A be CFFSG of X and B be CFFSG of U. Let x ∈ X and y ∈ U, then we have: h(A)(y) = {(y, h(KA)(y), h(LA)(y))}, is the image of A, where: h(K3 A) = { sup x∈h−1(y) K3 A(x) , h(x) = y 0 , otherwise. = ( sup x∈h−1(y) p3A(x))e 2πi( sup x∈h−1(y) ω3 A(x)) , h(x) = y 0 e2πi 0 , otherwise. h(L3 A) = { inf x∈h−1(y) L3 A(x) , h(x) = y 1 , otherwise. = ( inf x∈h−1(x) q3A(x))e 2πi( inf x∈h−1(y) ν3A(x)) , h(x) = y 1 , otherwise. And the set of pre-image of B is h−1(B)(x) = {(x, h−1(KB)(x), h −1(LB)(x))}, where: h−1(K3 B)(x) =(KB) 3(h(x)) = p3B(h(x))e 2πiω3 B(h(x)) h−1(L3 B)(x) =(LB) 3(h(x)) = q3B(h(x))e 2πiν3B(h(x)), ∀ x ∈ X. Lemma 1. Let h : X → U be a homomorphism from group X to group U, and let A be CFFSG of X, B be CFFSG of U. Then: 1) h(K3 A)(y) = h(p3A)(y)e 2πih(ω3 A)(y) ∀ y ∈ U. 2) h(L3 A)(y) = h(q3A)(y)e 2πif(ν3A)(y) ∀ y ∈ U. 3) h−1(K3 B)(x) = h−1(p3B)(x)e 2πih−1(ω3 B)(x) ∀ x ∈ X. 4) h−1(L3 B)(x) = h−1(q3B)(x)e 2πih−1(ν3B)(x) ∀ x ∈ X. Proof. 1) h(K3 A)(y) = sup x∈h−1(y) {K3 A(x); h(x) = y} = sup x∈h−1(y) {p3A(x)e2πiω 3 A(x); h(x) = y} E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 14 of 19 = sup x∈h−1(y) {p3A(x)}e 2πi sup x∈h−1(y) {ω3 A(x)} = h(p3A)(y)e 2πih(ω3 A)(y). 2) h(L3 A)(y) = inf x∈h−1(y) {L3 A(x); h(x) = y} = inf x∈h−1(y) {q3A(x)e2πiν 3 A(x); h(x) = y} = inf x∈h−1(y) {q3A(x)}e 2πi inf x∈h−1(y) {ν3A(x)} = h(q3A)(y)e 2πih(ν3A)(y). 3) h−1(K3 B)(x) = (KB) 3(h(x)) = p3B(h(x))e 2πω3 B(h(x)) = h−1(p3B)(x)e 2πih−1(ω3 B)(x) 4) h−1(L3 B)(x) = (LB) 3(h(x)) = q3B(h(x))e 2πν3B(h(x)) = h−1(q3B)(x)e 2πih−1(ν3B)(x) Example 5. Let (Z3,+3) and (Z,+) be complex fermatean fuzzy group (CFFG), where we define (Z3,+3) as in example 4. The map h : (Z,+) → (Z3,+3) is complex fermatean fuzzy homomorphism. Consider A = {1, 4, 5, 8, 11, 12} ⊆ Z, then h(A) = (x, h(KA)(x), h(LA)(x)). Then: 1) h(K3 A)(z) = sup x∈h−1(z) {K3 A(x); h(x) = z (mod 3) } = sup{K3 A(1),K3 A(4),K3 A(5),K3 A(8),K3 A(11),K3 A(12)} = sup{p3A(1)e2πiω 3 A(1), p3A(4)e 2πiω3 A(4), p3A(5)e 2πiω3 A(5), , p3A(8)e 2πiω3 A(8), p3A(11)e 2πiω3 A(11), p3A(12)e 2πiω3 A(12)} = sup{p3A(1), . . . , p3A(12)}e2πi sup{ω 3 A(1),...,ω3 A(12)} = sup{0.512, 0.027, 0.729}e2πi sup{0.512,0.216,0.343} = 0.729e2πi 0.512 2) h(L3 A)(z) = inf x∈h−1(z) {L3 A(x); h(x) = z (mod 3) } E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 15 of 19 = inf{L3 A(1),L3 A(4),L3 A(5),L3 A(8),L3 A(11),L3 A(12)} = inf{q3A(1)e2πiν 3 A(1), q3A(4)e 2πiν3A(4), q3A(5)e 2πiν3A(5), , q3A(8)e 2πiν3A(8), q3A(11)e 2πiν3A(11), q3A(12)e 2πiν3A(12)} = inf{q3A(1), . . . , q3A(12)}e2πi inf{ν 3 A(1),...,ν3A(12)} = inf{0.343, 0.125, 0.512}e2πi inf{0.216,0.512} = 0.125e2πi 0.216 Theorem 4. Let h : X epimorphism−−−−−−−−→ U, from (X, ∗1) to (U, ∗2), and let A be CFFSG of X. Then h(A) is CFFSG of U. Proof. Consider a two groups (X, ∗1) and (U, ∗2), with A = (KA,LA) is CFFSG, and want to show that h(A) = (h(KA), h(LA)) = (h(pA)(y)e 2πih(ωA)(y), h(qA)(y)e 2πih(νA)(y)) is CFFSG. At first, the set S1 = {(x, pA(x), qA(x)) : x ∈ X, 0 ≤ p3A(x) + q3A(x) ≤ 1} and S2 = {(x, ωA(x), νA(x)) : x ∈ X, 0 ≤ ω3 A(x) + ν3A(x) ≤ 1} are the amplitude and phase terms of CFFSG, since A is CFFSG and using Lemma 1. Then by Theorem[6.1] [35] and h is homomorphism, we have: i) a) h(p3A)(x1 ∗2 x2) = (h(pA)) 3(x1 ∗2 x2) ≥ (h(pA)) 3(x1) ∧ (h(pA)) 3(x2), b) h(q3A)(x1 ∗2 x2) = (h(qA)) 3(x1 ∗2 x2) ≤ (h(qA)) 3(x1) ∨ (h(qA)) 3(x2), c) h(ω3 A)(x1 ∗2 x2) = (h(ωA)) 3(x1 ∗2 x2) ≥ (h(ωA)) 3(x1) ∧ (h(ωA)) 3(x2), d) h(ν3A)(x1 ∗2 x2) = (h(νA)) 3(x1 ∗2 x2) ≤ (h(νA)) 3(x1) ∨ (h(νA)) 3(x2). ii) a) h(p3A)(x −1) = (h(pA)) 3(x−1) = (h(pA)) 3(x−1), b) h(q3A)(x −1) = (h(qA)) 3(x−1) = (h(qA)) 3(x−1), c) h(ω3 A)(x −1) = (h(ωA)) 3(x−1) = (h(ωA)) 3(x−1), d) h(ω3 A)(x −1) = (h(νA)) 3(x−1) = (h(νA)) 3(x−1). Consequently and by Lemma 1, we have: 1) (h(KA)) 3(x1 ∗2 x2) = h(K3 A)(x1 ∗2 x2) = h(p3A)(x1 ∗2 x2)e2πi h(ω3 A)(x1∗2x2) ≥ (h(p3A)(x1) ∧ h(p3A)(x2)) ∗ e2πi(h(ω 3 A)(x1)∧h(ω3 A)(x2)) = {h(p3A)(x1)e2πi h(ω3 A)(x1) ∧ h(p3A)(x2)e 2πi f(γ3 A)(x2)} = h(K3 A)(x1) ∧ h(K3 A)(x2) = (h(KA)) 3(x1) ∧ (h(KA)) 3(x2). 2) (h(LA)) 3(x1 ∗2 x2) = h(L3 A)(x1 ∗2 x2) = h(q3A)(x1 ∗2 x2)e2πi f(ν3A)(x1∗2x2) ≤ (h(q3A)(x1) ∨ h(q3A)(x2)) ∗ e2πi(h(ν 3 A)(x1)∨h(ν3A)(x2)) = {h(q3A)(x1)e2πi h(ν3A)(x1) ∨ h(q3A)(x2)e 2πi h(ν3A)(x2)} = h(L3 A)(x1) ∨ h(L3 A)(x2) = (h(LA)) 3(x1) ∨ (h(LA)) 3(x2). 3) (h(KA)) 3(x−1) = h(K3 A)(x −1) = h(p3A)(x −1)e2πi h(ω3 A)(x−1) = h(p3A)(x)e 2πi h(ω3 A)(x) = h(K3 A)(x) = (h(KA)) 3(x). 4) (h(LA)) 3(x−1) = h(L3 A)(x −1) = E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 16 of 19 h(q3A)(x −1)e2πi h(ν3A)(x−1) = h(q3A)(x)e 2πi f(ν3A)(x) = h(L3 A)(x) = (h(LA)) 3(x). Hence result follows. Theorem 5. Let f : X isomorphism−−−−−−−−→ U, from (X, ∗1) to (U, ∗2), and let B be CFFSG of U. Then h−1(B) is CFFSG of X. Proof. The proof will be similar to previous theorem, and that by using Lemma 1 with Theorem[6.2] [35]. Theorem 6. Let h : X epimorphism−−−−−−−−→ U, from (X, ∗1) to (U, ∗2), and let A be CFFNSG of X. Then h(A) is CFFNSG of U. Proof. According to Proposition 5 with Theorem[6.3] [35], and inspiring the proof of Theorem 4, result will follows. Theorem 7. Let h : X isomorphism−−−−−−−−→ U, from (X, ∗1) to (U, ∗2), and let B be CFFNSG of U. Then h−1(B) is CFFNSG of X. Proof. According to Proposition 5 with Theorem[6.4] [35], and inspiring the proof of Theorem 4, result will follows. Example 6. Let (X, ∗1) = K4 =< a, b|a2 = b2 = (ab)2 = 1 >, i.e. the Klein four- group, and (U, ∗2) = D2 ∼= Z2 ⊕ Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. In addition, for B =< ℓ, p(ℓ)e2πiω(ℓ), q(ℓ)e2πiν(ℓ) > that defined a CFFSG on D2, see Example 1. Then B defined CFFNSG of D2, where the dihedral group D2 is an abelian group. Hence, we can find homomorphism function that is bijection; h : K4 isomorphism−−−−−−−−→ D2, with h(1) = (0, 0), h(a) = (1, 0), h(b) = (0, 1), h(ab) = (1, 1). Now, according to definition 12 h−1(K3 B)(u) = p3B(h(u))e 2πiω3 B(h(u)) and h−1(L3 B)(u) = q3B(h(u))e 2πiν3B(h(u)), ∀ u ∈ K4. Hence, by this definition we get that h−1(B) is CFFNSG of K4. 6. Conclusion This research provides a theoretical foundation for the complex Fermatean fuzzy sub- group (CFFSG) and examines its algebraic properties. The concepts of complex Fermatean fuzzy normal subgroups and complex Fermatean fuzzy cosets were introduced. Addition- ally, the conditions under which a complex Fermatean fuzzy subgroup can be a complex Fermatean fuzzy normal subgroup were explored. A homomorphism between two com- plex Fermatean fuzzy subgroups and its properties were also discussed. As a direction for future research, we plan to refine the definition of CFFSG by replacing the minimum and maximum operations with T-norm and S-norm functions, respectively. Furthermore, fixed-point theory could be integrated and extended within the context of CFFSG. The approach presented here can be progressively applied to other algebraic structures, such E.A. AbuHijleh, A. Alkouri / Eur. J. Pure Appl. Math, 18 (1) (2025), 5572 17 of 19 as integral domains, fields, rings, and factor groups. By incorporating periodic informa- tion into the CFFSG framework, the current structure could facilitate the development of cryptographic primitives and be applied to the generalization of new algorithms. An- other avenue for future work is upgrading CFFSG to the complex q-rung orthopair fuzzy subgroup. References [1] I. Abu-Irwaq, W. Shatanawi, A. Bataihah, and I. Nuseir. Fixed point results for nonlinear contractions with generalized ω-distance mappings. UPB Sci. Bull. Ser. 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