On the functional equation ${\varvec{f}}({\varvec{x}})+{\varvec{f}}({\varvec{y}})=\mathbf{max} \{{\varvec{f}}({\varvec{xy}}),{\varvec{f}}({\varvec{xy}}^{-{\varvec{1}}})\}$ on groups

Abstract

We analyse the functional equation

$$\begin{aligned} f(x)+f(y)=\max \{f(xy),f(xy^{-1})\} \end{aligned}$$

for a function $f:G\rightarrow \mathbb R$ where G is a group. Without further assumption it characterises the absolute value of additive functions. In addition $\{z\in G\mid f(z)=0\}$ is a normal subgroup of G with abelian factor group.