Abstract: Representation theory (G-module theory) has had its origin in the 20thcentury. In the 19th century, groups were generally regarded as subsets of some permutation set, or of the set GL(V) of automorphisms of a vector space V, closed under composition and inverse. Here we consider , the periodic arithmetical functions (mod r), define a fuzzy G-module on it and verify the quasi injective property of its summands.