The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results

Abstract

This research paper focuses on the investigation of the solutions χ:G→R of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)χ(y), for every x,y∈G, where G is any group. We determine that if a group G is divisible by two and three, then every non-zero solution is necessarily strictly positive; by the work of Toborg, we can then conclude that the solutions are exactly the e|α| for an additive function α:G→R. Moreover, our investigation yields reliable solutions to a functional equation on any group G, instead of being divisible by two and three. We also prove the existence of normal subgroups Zχ and Nχ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G/Nχ.