群的概念教学中几个有限生成群的例子

Several Examples of Finitely Generated Groups in the Concept Teaching of Groups

群的概念是抽象代数中的最基本的概念之一,在抽象代数课程的教学环节中融入一些有趣的群例,借助于这些较为具体的群例来解释抽象的群理论,对于激发学生的学习兴趣以及锻炼学生的数学思维能力等方面都会起到一定的积极作用。该文介绍了一种利用英文字母表在一定的规则下构造的有限生成自由群的例子,即该自由群的同音商,称为英语同音群。此外,该文结合线性代数中的矩阵相关知识,给出了有限生成群SL;(Z)以及若于有限生成特殊射影线性群的例子。

The concept of group is one of the most basic concepts in abstract algebra. Integrating some interesting group examples into the teaching of abstract algebra course and explaining the abstract group theory with the help of these more specific group examples will play a positive role in stimulating students’ learning interest and training students’ mathematical thinking ability. In this paper, we introduce an example of finitely generated free group by using the English alphabet under some certain rules, which is called homophonic quotients of free groups, or briefly called English homophonic group. In addition, combined with the theory of matrix in linear algebra, we give some examples of about finitely generated group SL;(Z) and finitely generated special projective linear groups.