On the fuzzification of Lagrange's theorem in $ (\alpha, \beta) $-Pythagorean fuzzy environment

Abstract

<abstract> 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. </abstract>