Algebraic structure of some complex intuitionistic fuzzy subgroups and their homomorphism

The complex fuzzy environment is an innovative tool to deal with fuzzy situations in different mathematical problems. Aiming at the concept of complex intuitionistic fuzzy subgroups, this paper has introduced cut-subsets of complex intuitionistic fuzzy sets, and studied the relationship among the cut-subsets and complex intuitionistic fuzzy subgroups, complex intuitionistic fuzzy Abel subgroups, and complex intuitionistic fuzzy cyclic subgroups. Further, we gave the left and right cosets of complex intuitionistic fuzzy subgroups, defined complex intuitionistic fuzzy normal subgroups, and discussed some of their algebraic properties. Based on this thought, we proposed a new concept of $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy subgroups, and proved that an $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. At the same time, $({\alpha }_{1,2},{\beta }_{1,2})$-complex intuitionistic fuzzy normal subgroups and their cosets were introduced. Finally, we established a general homomorphism of complex intuitionistic fuzzy subgroups, and studied the relationship between the image and pre-image of complex intuitionistic fuzzy subgroups and complex intuitionistic fuzzy normal subgroups, respectively, under group homomorphism.