A Certain Exact Sequence

Summary:Let $G$ be a group and $H$ an abelian group. Let $J^{*} (G, H)$ be the set of solutions $f \colon G \to H$ of the Jensen functional equation $f(xy)+f(xy^{-1}) = 2f(x)$ satisfying the condition $f(xyz) - f(xzy) = f(yz)-f(zy)$ for all $x, y , z \in G$. Let $Q^{*} (G, H)$ be the set of solutions $f \colon G \to H$ of the quadratic equation $f(xy)+f(xy^{-1}) = 2f(x) + 2f(y)$ satisfying the Kannappan condition $f(xyz) = f(xzy)$ for all $x, y, z \in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f \colon G \to H$ of the Whitehead equation is of the form $4f = 2 \varphi + 2 \psi $, where $2\varphi \in J^* (G, H)$ and $2\psi \in Q^* (G, H)$. Moreover, if $H$ has the additional property that $2h = 0$ implies $h = 0$ for all $h \in H$, then every solution $f \colon G \to H$ of the Whitehead equation is of the form $2f = \varphi + \psi $, where $\varphi \in J^*(G,H)$ and $2\psi (x) = B(x, x)$ for some symmetric bihomomorphism $B \colon G \times G \to H$