On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups

In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (<inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG) and prove that every IVFSG is <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG. We use this ideology to define the concepts of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF cosets, <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFSG and investigate the condition under which a <inline-formula> <tex-math notation="LaTeX">$\mathrm {\Gamma }$ </tex-math></inline-formula>-IVFS is <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> relative to the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFNSG and acquire a correspondence between each <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of a group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVF(N)SG of its quotient group. Furthermore, we define the index of <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG and establish the <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-interval valued fuzzification of Lagrange’s theorem of any <inline-formula> <tex-math notation="LaTeX">$\Gamma $ </tex-math></inline-formula>-IVFSG of a finite group <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>.