A composite functional equation from algebraic aspect

Abstract

In this paper we discuss the composite functional equation $$f(x+2f(y)) = f(x)+y+f(y)$$ on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers.