﻿ 正多面体自同构群的一点注记 A Note on the Automorphism Groups of Regular Polyhedrons

Pure Mathematics
Vol.07 No.03(2017), Article ID:20567,7 pages
10.12677/PM.2017.73024

A Note on the Automorphism Groups of Regular Polyhedrons

Lu Yu, Qi Cai

School of Mathematics, Yunnan Normal University Yunnan, Kunming Yunnan

Received: Apr. 30th, 2017; accepted: May 15th, 2017; published: May 19th, 2017

ABSTRACT

In this paper, by using the elementary method of permutation group theory, we give a simple characterization of the full automorphism groups of regular polyhedrons.

Keywords:Regular Polyhedron, Permutation Groups, Full Automorphism Groups

Copyright © 2017 by authors and Hans Publishers Inc.

1. 引言

Figure 1. The convex polyhedras

Figure 2. The Kepler-Poinsot polyhedra

2. 预备知识

Figure 3. The regular hexahedron and the regular octahedron are dual

Figure 4. The regular dodecahedron and the regular icosahedron are dual

3. 定理的证明

3.1. 正四面体

，有

3.2. 正六面体

Figure 5. Regular tetrahedron

Figure 6. Regular hexahedron

，有

3.3. 正十二面体

：上、下顶点中心为轴旋转72˚，：以所在平面为中心作反射，：以所在平面作反射，其相应的置换表示如下：

，有：

Figure 7. The planar gragh of regular dodecahedron

，于是有

，此时

A Note on the Automorphism Groups of Regular Polyhedrons[J]. 理论数学, 2017, 07(03): 186-192. http://dx.doi.org/10.12677/PM.2017.73024

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