Parter, Periodic and Coperiodic Functions on Groups and their Characterization

Decomposer functional equations were introduced by the author and have been completely solved on arbitrary groups. Their solutions are as decomposer functions and play important role regarding to decomposition (factorization) of groups by their two subsets. In this paper, we introduce an important class of strong decomposer functions, namely parter (or cyclic decomposer) functions. As some important applications of this topic, we characterize all periodic , coperiodic functions in arbitrary groups and give general solution of their functional equations: f(bx) = f(x) , f(xb) = f(x), f(bx) = bf(x) and f(xb) = f(x)b. Moreover, we characterize all parter functions in arbitrary groups and completely solve the decomposer equation with the condition which its -range is a cyclic subgroup of G. Finally, we give some functional characterization for related projections and b-parts functions and also, we introduce some uniqueness conditions for b-parts of real numbers.