循环群的一些讨论

Some Discussions on Cyclic Group

循环群是一类很重要的群。根据生成元阶的情况,循环群分成两类:有限循环群和无限循环群。对于这两类循环群,主要分析生成元是否具有唯一性以及循环群子群的一些性质,利用同态和同构建立起循环群之间的关系,并推论出几个最基本的判断循环群的方法。

The cyclic group is a kind of inaportant group. According to the differences of the generator, the cyclic group is divided into two kinds： finite cyclic group and infinite cyclic group. For the two kinds of cyclic group, the uniqueness of the generator and some natures of subgroup were, discussed. The relations be- tween the cyclic groups by the homomorphism and isomorphism were established. Some methods are posed on the judgments of cyclic group at the end of the article.