Mathematics ( IF 1.747 ) Pub Date : 2021-02-14 , DOI: 10.3390/math9040382
Muhammad Sarfraz; Qi Liu; Yongjin Li

This research paper focuses on the investigation of the solutions $\chi :G\to \mathbb{R}$ of the maximum functional equation $max\left\{\phantom{\rule{0.166667em}{0ex}}\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\phantom{\rule{0.166667em}{0ex}}\right\}=\chi \left(x\right)\chi \left(y\right),$ for every $x,y\in G,$ 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 ${e}^{|\alpha |}$ for an additive function $\alpha :G\to \mathbb{R}$. 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 ${Z}_{\chi }$ and ${N}_{\chi }$ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group $G/{N}_{\chi }$.

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