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In this paper we study some combinatorial properties of biquadratic irrational number field Q (i , 3) , under the action of a Bianchi group Γ 3 = P S L (2 , O 3 ). In this experiment it is revealed that a special class of elements exists; that is, for an element ξ its conjugate ξ ¯ has different signs in the closed path (orbits) for the action of Γ 3 over Q (i , 3) , known as ambiguous numbers. It is also proved that the orbit Γ ξ defined on a finite number of ambiguous numbers succeeding a unique closed path. [ABSTRACT FROM AUTHOR]

In this study, we presents the idea of η -intuitionistic fuzzy subgroup (IFSG) defined on η -intuitionistic fuzzy set (IFS). Furthermore, we prove that every IFSG is an η -IFSG. Also, we extend the study of this notion to define η -intuitionistic fuzzy cosets and η -intuitionistic fuzzy normal subgroups of a given group and investigate some of their fundamental algebraic features. Besides, we define the η -intuitionistic fuzzy homomorphism between two η -IFSG's and show that an η -intuitionistic fuzzy homomorphic image (inverse image) of the η -IFSG is an η -IFSG. [ABSTRACT FROM AUTHOR]