A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of [Formula: see text]-complex fuzzy sets and then define [Formula: see text]-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an [Formula: see text]-complex fuzzy subgroup and define [Formula: see text]-complex fuzzy normal subgroups of given group. We extend this ideology to define [Formula: see text]-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that [Formula: see text]-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of [Formula: see text]-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the [Formula: see text]-complex fuzzy subgroup of the classical quotient group and show that the set of all [Formula: see text]-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of [Formula: see text]-complex fuzzy subgroups and investigate the [Formula: see text]-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.