| Summary: | In this study, the <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy normalizer and centralizer of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> intuitionistic fuzzy subgroup are proposed. The <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy centralizer is normal subgroup of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy subgroup of Abelian (cyclic) group is <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and pre-image of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy Abelian (cyclic) subgroup are <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism.
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