Functional equations on groups

Summary

A functional equation on a group is a functional equation whose unknown functions are defined on a group. 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. 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 and representations of the group, which links the subject to harmonic analysis.