---
tag: RJYSXGZTTE
kind: concept
name: Functional equations on groups
msc: [39B52, 43A40]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Group_homomorphism
---

## Summary

A functional equation on a group is a [[functional equation|SRAN4EKWS0]] whose unknown functions are defined on a [[group|G5S1M9]].
The arguments of these functions are formed from the variables by the group operation and inversion.
The basic example is Cauchy's equation $f(xy) = f(x) f(y)$ for maps $f$ from $G$ to another group, whose solutions are exactly the [[group homomorphisms|0JYTG5K9Y4]].
Classical equations on $\R$, such as the d'Alembert, Jensen and quadratic equations, have analogues on arbitrary groups.
Their solutions are often described by homomorphisms, [[characters|H1WNVN5V7F]] and representations of the group, which links the subject to harmonic analysis.

## Explanations

### L1 human

Many functional equations on the real line only use addition and subtraction of the variables.
Any group has an operation and inverses, so the same equations make sense on any group.
For example, on the group of nonzero real numbers under multiplication, $f(xy) = f(x) + f(y)$ is solved by $f(x) = \ln \abs{x}$.
On non-commutative groups, such as groups of matrices, new phenomena appear, because $xy$ and $yx$ may differ.