Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in numerous papers. Among them are absolute maximum nonlinear functions on finite nonabelian groups introduced by Poinsot and Pott [15] in 2011. Recently, properties and constructions of absolute maximum nonlinear functions were studied in [19]. In this paper we study the characterizations of absolute maximum nonlinear functions on arbitrary finite groups. Then as an application of these characterizations, we discuss the existence of absolute maximum nonlinear functions on dihedral groups. We will prove that for a dihedral group $D_{2n}$ of order 2n, where $n\ge 3$, if there is an absolute maximum nonlinear function on $D_{2n}$, then $n\in \{3,12,15,18,30,33,42,66,138\}$. In particular, if there exists an absolute maximum nonlinear function from $D_{2n}$ to ${\mathbb{Z}}_2$, where ${\mathbb{Z}}_2$ is the group of order 2, then we show that $n=12$. All absolute maximum nonlinear functions from $D_{24}$ to ${\mathbb{Z}}_2$ will be determined.

许多论文已经研究了有限域和有限（阿贝尔或非阿贝尔）群上的高度非线性函数（完美非线性、最大非线性等）。其中包括 Poinsot 和 Pott [15] 在 2011 年引入的有限非阿贝尔群上的绝对最大非线性函数。最近，[19] 研究了绝对最大非线性函数的性质和构造。在本文中，我们研究了任意有限群上的绝对最大非线性函数的表征。然后作为这些特征的应用，我们讨论了二面体群上绝对最大非线性函数的存在。我们将证明对于二面体群$D_{2n}$2 n阶，其中$n\ge 3$, 如果存在一个绝对最大非线性函数 $D_{2n}$， 然后 $n\in \{3,12,15,18,30,33,42,66,138\}$. 特别地，如果存在一个绝对最大非线性函数$D_{2n}$ 到 ${\mathbb{Z}}_2$， 在哪里 ${\mathbb{Z}}_2$ 是 2 阶群，那么我们证明 $n=12$. 所有绝对最大非线性函数来自$D_{24}$ 到 ${\mathbb{Z}}_2$ 将被确定。