This research paper focuses on the investigation of the solutions $\chi :G\to \mathbb{R}$ of the maximum functional equation $max\{\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\}=\chi \left(x\right)\chi \left(y\right),$ 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 $e^{\left|\alpha \right|}$ 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 $G/N_{\chi }$.

中文翻译：

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群和相关结果上的函数方程max {χ（xy），χ（xy-1）} =χ（x）χ（y）

本研究论文着重于解决方案的研究 $\chi ：G\to \mathbb{[R}$ 泛函方程 $最大限度\{\chi （Xÿ），\chi （X{ÿ}^{-\mathrm{1个}}）\}=\chi （X）\chi （ÿ），$ 每一个 $X，ÿ\in G，$其中G是任何组。我们确定如果组G可被2和3整除，则每个非零解都必须严格为正；通过Toborg的工作，我们可以得出结论，解决方案正是${Ë}^{\left|\alpha \right|}$ 用于加法功能 $\alpha ：G\to \mathbb{[R}$。此外，我们的研究为任何G组的函数方程提供了可靠的解，而不是被二和三整除。我们还证明了正常亚群的存在${ž}_{\chi }$ 和 ${ñ}_{\chi }$满足某些性质的任意组G的任意一个，并且任何解都可以解释为阿贝尔因子组的函数$G/{ñ}_{\chi }$。