Some Discussions on Cyclic Group

The cyclic group is a kind of important 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 between 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.