Jensen's Functional Equation on Involution-Generated Groups: An ($\mathrm{SR}_2$) Criterion and Applications

Abstract

We study the Jensen functional equations on a group $G$ with values in an abelian group $H$: \begin{align} \tag{J1}\label{eq:J1} f(xy)+f(xy^{-1})&=2f(x)\qquad(\forall\,x,y\in G),\\ \tag{J2}\label{eq:J2} f(xy)+f(x^{-1}y)&=2f(y)\qquad(\forall\,x,y\in G), \end{align} with the normalization $f(e)=0.$ Building on techniques for the symmetric groups $S_n$, we isolate a structural criterion on $G$ -- phrased purely in terms of involutions and square roots -- under which every solution to \eqref{eq:J1} must also satisfy \eqref{eq:J2} and is automatically a group homomorphism. Our new criterion, denoted $(\mathrm{SR}_2)$, implies that $S_1(G,H) = S_{1,2}(G,H) = \mathrm{Hom}(G,H)$, applies to many reflection-generated groups and, in particular, recovers the full solution on $S_n.$ Furthermore, we give a transparent description of the solution space in terms of the abelianization $G/[G,G],$ and we treat dihedral groups $D_m$ in detail, separating the cases $m$ odd and even. The approach is independent of division by 2 in $H$ and complements the classical complex-valued theory that reduces \eqref{eq:J1} to functions on $G/[G,[G,G]].$