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

This research paper focuses on the investigation of the solutions χ : G → R of the maximum functional equation max { χ ( x y ) , χ ( x y − 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 Z χ and N χ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G / N χ .