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 ) + 2 g ( x ) + 2 g ( 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 ( x 2 ) = f ( x ) + f ( y ) + f ( x 2 ) 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.