A Short Proof of Alienation of Additivity from Quadraticity

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)f\left( {x + y} \right) + g\left( {x + y} \right) + g\left( {x - y} \right) = 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(x2)f\left( {x + y} \right) + f({x^2}) = 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.