A cyclic group is a mathematical

group which is generated by one of its

elements, i.e. every element x can be written as x = a

^{k}, where a is the generator and k is an

integer.

Cyclic groups are important in number theory because any cyclic group of infinite order is isomorphic to the group formed by the set of all integers and addition as the operation, and any finite cyclic group of order n is isomorphic to the set of all integers modulo n, using addition followed by taking the modulo as the operation.