Introduction to cyclic groups

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:

  1. Every cyclic group is abelian, but abelian groups don't need to be cyclic
  2. Every subgroup of a cyclic group is cyclic but the group is not necessarily cyclic if all its subgroups are cyclic
  3. The number of generators of a finite cyclic group of order is , where is the number of integers less than and coprime to
  4. If a cyclic group is generated by an element of order , then is a generator of if and only if
  5. Every group of order , with prime, is cyclic
  6. A cyclic group of finite order is isomorphic to the additive group of residue classes modulo