A cyclic group is a group that can be generated by just one element .
If a group contains an element such that every element is of the form , where , then is said to be a cyclic group and is generated by . In this case we write:
If is a generator of , then is also a generator of .
But let's make an example: the multiplicative group is a cyclic group of order with generator , i.e.
Some of the most fundamental properties of cyclic groups are listed below:
- Every cyclic group is abelian, but abelian groups don't need to be cyclic
- Every subgroup of a cyclic group is cyclic but the group is not necessarily cyclic if all its subgroups are cyclic
- The number of generators of a finite cyclic group of order is , where is the number of integers less than and coprime to
- If a cyclic group is generated by an element of order , then is a generator of if and only if
- Every group of order , with prime, is cyclic
- A cyclic group of finite order is isomorphic to the additive group of residue classes modulo