A short proof of alienation of additivity from quadraticity

Abstract

Without the use of pexiderized versions of abstract polynomials theory, we show that on 2-divisible groups the functional equation f(x + y) + g(x + y) + g(x − y) = f(x) + f(y)+ 2g(x) + 2g(y) forces the unknown functions f and g to be additive and quadratic, respectively, modulo a constant. Motivated by the observation that the equation f(x + y) + f(x2) = f(x) + f(y)+ f(x)2 implies both the additivity and multiplicativity of f, we deal also with the alienation phenomenon of equations in a single and several variables.