In fact, it is the only infinite cyclic group up to isomorphism. Notice that a cyclic group can have more than one generator. If n is a positive integer, is a cyclic group of order n generated by 1. For example, 1 generates , since In other words, if you add 1 to itself repeatedly, you eventually cycle back to 0. Notice that 3 also generates :
In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.