---
tag: XZY0H7K195
kind: definition
name: Functional equations on groups
msc: [39B52, 43A40]
defines: RJYSXGZTTE
uses: [SRAN4EKWS0, G5S1M9, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Group_homomorphism
---

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$, and let $Y$ be a set.
A *functional equation on $G$* is a [[functional equation|SRAN4EKWS0]] for one or more unknown [[functions|0EPAT51DXQ]] $f_1, \dots, f_m \colon G \to Y$ in which:

1. the variables $x_1, \dots, x_k$ range over all of $G$;
2. each unknown function is evaluated at words in $x_1, \dots, x_k$, their inverses and fixed elements of $G$;
3. the two sides of the equation are combined from these values by fixed operations on $Y$.

A solution is a tuple $(f_1, \dots, f_m)$ for which the equation holds for all $x_1, \dots, x_k \in G$.

## Remarks

Example: for $G$ a group and $Y = \C$, the equation $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$ is d'Alembert's equation on $G$.
It has the solution $f(x) = (\chi(x) + \chi(x)^{-1})/2$ for every homomorphism $\chi \colon G \to \C^{\times}$, as a direct expansion shows.
Non-example: the equation $f(f(x)) = x$ for $f \colon G \to G$ does not fit the definition.
There the unknown function is evaluated at a value of itself, not at a word in the variables, so it is an iterative equation.
For $G = (\R, +)$, this setting includes the classical equations of Cauchy, d'Alembert and Pexider, and Jensen's equation in the equivalent form $f(x + y) + f(x - y) = 2 f(x)$.
Solutions on $G$ are often required to be continuous or measurable when $G$ is a topological group.