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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

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Aloizio Macedo May 12 '19 at 20:56
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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 $S_3$.
  • unlike cyclic groups, periodic groups need not be countable, in fact they can be arbitrarly big. Consider any set $X$ and the set of all functions $Func(X,\mathbb{Z}_2)$ with pointwise addition and note that it is periodic with cardinality $2^{|X|}$.
  • as a consequence: the class of all periodic groups (up to isomorphism) is not a set, unlike cyclic groups. In a sense there are lots of periodic groups.

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.

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    $\begingroup$ I think originally "cyclic group" meant finite cyclic group, for the obvious intuition, and that it has been progressively been found practical to also call the infinite "monogenic" group cyclic. Similarly a permutation of a finite set splits into finite cycles, and it's been practical to also allow infinite "cycles" in permutations of infinite sets, although these are points which precisely do not "cycle" in the "real world" meaning. $\endgroup$ – YCor May 12 '19 at 10:04
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    $\begingroup$ By the way I always found the terminology of periodic groups counterintuitive, since precisely there is no period. I prefer call them torsion groups. $\endgroup$ – YCor May 12 '19 at 10:05

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