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

Dang Vo Phuc

TL;DR

This work addresses Jensen-type equations on noncommutative groups by introducing a structural square-root criterion $SR_2$ based on involutions. It proves that if $G$ satisfies $SR_2$, every normalized solution to $(J1)$ is a group homomorphism and automatically satisfies $(J2)$, without requiring divisibility by 2 in the codomain $H$. Consequently the solution space collapses to $Hom(G_{ab},H)$, and a parity description emerges when $G$ is generated by involutions. The authors apply this to symmetric and dihedral groups, yielding a unified view and a sharp odd/even dichotomy for dihedral groups, with explicit counterexamples in the even case, and discuss broader Coxeter-type implications. The results complement the complex-valued theory by avoiding division by 2 and providing a purely group-theoretic, endomorphism-free framework.

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 $G/[G,[G,G]].$

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

TL;DR

This work addresses Jensen-type equations on noncommutative groups by introducing a structural square-root criterion based on involutions. It proves that if satisfies , every normalized solution to is a group homomorphism and automatically satisfies , without requiring divisibility by 2 in the codomain . Consequently the solution space collapses to , and a parity description emerges when is generated by involutions. The authors apply this to symmetric and dihedral groups, yielding a unified view and a sharp odd/even dichotomy for dihedral groups, with explicit counterexamples in the even case, and discuss broader Coxeter-type implications. The results complement the complex-valued theory by avoiding division by 2 and providing a purely group-theoretic, endomorphism-free framework.

Abstract

We study the Jensen functional equations on a group with values in an abelian group : \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 Building on techniques for the symmetric groups , we isolate a structural criterion on -- 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 , implies that , applies to many reflection-generated groups and, in particular, recovers the full solution on Furthermore, we give a transparent description of the solution space in terms of the abelianization and we treat dihedral groups in detail, separating the cases odd and even. The approach is independent of division by 2 in and complements the classical complex-valued theory that reduces \eqref{eq:J1} to functions on

Paper Structure

This paper contains 7 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. A structural square-root criterion
  3. Applications
  4. Symmetric groups
  5. Dihedral groups
  6. Consequences and examples
  7. Final remark: Comparison with complex-valued theory and semigroup variants

Key Result

Lemma 2.3

Let $G$ be a group and $H$ an abelian group. Suppose $f:G\to H$ satisfies the two Jensen identity eq:J1 with $f(e)=0$. Then for all $x,y,z\in G$:

Theorems & Definitions (24)

  • Definition 2.1: The $(\mathrm{SR}_2)$ property
  • Remark 2.2
  • Lemma 2.3: Basic identities without dividing by $2$
  • proof
  • Remark 2.4
  • Theorem 2.5: Two involutions: torsion bounds and square roots
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7: Reordering invariance under $2f\equiv 0$
  • ...and 14 more