On the fuzzification of Lagrange's theorem in (α, β)-Pythagorean fuzzy environment

An (alpha, beta)-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the (alpha, beta)-Pythagorean fuzzy order of elements of groups in (alpha, beta)-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between (alpha, beta)-Pythagorean fuzzy order of an element of a group in (alpha, beta)-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for (alpha, beta)-Pythagorean fuzzy sets is introduced. The concept of (alpha, beta)-Pythagorean fuzzy normalizer and (alpha, beta)-Pythagorean fuzzy centralizer of (alpha, beta)-Pythagorean fuzzy subgroups are developed. Further, (alpha, beta)-Pythagorean fuzzy quotient group of an (alpha, beta)-Pythagorean fuzzy subgroup is defined. Finally, an (alpha, beta)-Pythagorean fuzzy version of Lagrange's theorem is proved.

Automated summary

（English Summary:） This paper introduces the concept of (alpha, beta)-Pythagorean fuzzy order of elements, including relative subgroup, normalizer, centralizer, and a fuzzy version of Lagrange's theorem, providing an efficient tool for handling vagueness in a fuzzy environment.