二面体群与循环群的幂映射的有向图

Digraph from Power Mapping on Dihedral Groups and Cyclic Groups

摘要: 利用$f\left(x\right)=x^k$ 定义幂映射$f:G\to G$ ($G=D_{2n}⋊C_m$ )，其中$x\in G$ ，$k$ 是自然数。若把$G$ 作为顶点集，$v$ 和$u$ 有边连接当且仅当$u=v^k$ ，则上面定义的映射构成幂映射图。本文研究了二面体群与循环群半直积上的幂映射图的结构与性质，包括圈顶点的存在性，以及圈数、圈长度和顶点入度等。

Abstract: Define the power mapping $f:G\to G$ ($G=D_{2n}⋊C_m$ ) by $f\left(x\right)=x^k$ , for $x\in G$ , and $k$ is a positive integer. The k-power digraph consists of vertex set $G$ and there is a directed edge from $v$ to $u$ if and only if $u=v^k$ . This paper studies the structure and properties of the power mapping graph on the semidirect product of a dihedral group and a cyclic group, including the existence of cyclic vertices, as well as the number of cycles, cycle lengths, and in-degrees of vertices.