(α1, 2, β1, 2)-complex intuitionistic fuzzy subgroups and its algebraic structure

A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of ( ${\alpha }_{1,2},{\beta }_{1,2}$) complex intuitionistic fuzzy sets and ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under ( ${\alpha }_{1,2},{\beta }_{1,2}$)-complex intuitionistic fuzzy subgroups.