I have read that a cyclic group G is one that can be generated by a single element a called the generator , aϵG.
While looking up Wikipedia for Torsion Groups(periodic groups), I found:
"In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group."
I am confused, after this I couldn't find a satisfying difference between the two(periodic and cyclic groups).
Thanks
Best Answer
Cyclic group is a group with single generator. These are all well known. Up to isomorphism a cyclic group is either $\mathbb{Z}$ or its quotient $\mathbb{Z}_n:=\mathbb{Z}/(n)$.
Now a group $G$ is periodic if every element is of finite order. So this includes all finite groups, but not only.
Few facts:
I understand that words "cyclic" and "periodic" are confusing because in real world they pretty much mean the same thing. Unfortunately in mathematics these notions are completely different. All you can do is just learn that.