What is the difference between cyclic groups and periodic groups?

Question

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

Answer 1

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:

• a cyclic group is periodic if and only if it is finite, i.e. $\mathbb{Z}$ is the only cyclic group that is not periodic
• since every finite group is periodic then periodic groups need not be cyclic. The simpliest example is $\mathbb{Z}_2\oplus\mathbb{Z}_2$.
• unlike cyclic groups, periodic groups need not be abelian. Consider $S_3$.
• periodic groups need not be finite. Consider the infinite direct sum $\bigoplus_{i=1}^{\infty}\mathbb{Z}_2$, i.e. all sequences with values in $\mathbb{Z}_2$ such that almost every element of the sequence is $0$ (with pointwise addition).

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.