Parter, Periodic and Coperiodic Functions on Groups and their Characterization

Decomposer functional equations were introduced by theauthor and have been completely solved on arbitrary groups. their solutionsare as decomposer functions and play important role regardingto decomposition (factorization) of groups by their two subsets. in thispaper, we introduce an important class of strong decomposer functions,namely parter (or cyclic decomposer) functions. as some important applicationsof this topic, we characterize all periodic , coperiodic functionsin 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 andcompletely solve the decomposer equation with the condition which its-range is a cyclic subgroup of g. finally, we give some functional characterizationfor related projections and b-parts functions and also, weintroduce some uniqueness conditions for b-parts of real numbers.