Jensen's Functional Equation on Involution-Generated Groups: A Square-Root Criterion, Rigidity Phenomena, and Obstruction Spaces

Abstract

Let $G$ be a group and $H$ an abelian group. We study normalized solutions $f:G\to H$ of the Jensen equation $f(xy)+f(xy^{-1})=2f(x)$ and its companion $f(xy)+f(x^{-1}y)=2f(y)$. Removing standard 2-divisibility constraints on $H$, we shift the analytical focus to the domain $G$. We introduce the square-root criterion $(\mathrm{SR}_2)$ for groups generated by involutions: if every product of two involution generators admits a square root in $G$, then every normalized Jensen solution is a group homomorphism, yielding the exact identification $S_1(G,H) = \mathrm{Hom}(G,H) \cong \mathrm{Hom}(G_{\mathrm{ab}},H)$. We apply this intrinsic mechanism to various algebraic and geometric structures. For arbitrary semidirect products $A\rtimes_\sigma C_2$, the existence of an $(\mathrm{SR}_2)$ involution generating set exhibits a strong rigidity phenomenon, forcing $\sigma$ to be inversion and the square map on $A$ to be surjective. When $(\mathrm{SR}_2)$ fails, we develop an exact algebraic theory of the obstruction space $\mathcal{O}(G,H) := S_1(G,H)/\mathrm{Hom}(G,H)$. By identifying the canonical Jensen equivalence relation with the maximal elementary abelian 2-quotient $G/G^{(2)}$, we explicitly compute this defect, determining its exact dimension for finite cases and even dihedral groups. Finally, extending beyond discrete settings, we prove that the real orthogonal group $O(n)$ generated by hyperplane reflections satisfies $(\mathrm{SR}_2)$. Consequently, without assuming any analytical regularity, every normalized Jensen solution on $O(n)$ is a homomorphism, taking values strictly in $H[2]$.