A short proof of alienation of additivity from quadraticity

Abstract

Without a 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.
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.