# Is there an infinite group whose elements all have finite order? [closed]

Is there an infinite (edit: but finitely generated) group G such that for all g in G, |g| is finite?

If so, how many such groups exist?

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## closed as too localized by Ryan Budney, J.C. Ottem, Ian Agol, Felipe Voloch, Pete L. ClarkMar 5 '11 at 21:12

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The direct sum of infinitely many copies of $\mathbb Z_2$. –  Ryan Budney Mar 5 '11 at 20:12
Edit to specify "infinite finitely generated group"? Anyone? –  Elizabeth S. Q. Goodman Mar 5 '11 at 20:24
Yes, Grigorchuk group is a finitely generated infinite 2-group. I.e, for each $g\in G$ there is an $n$ such that g^(2^n)=1. –  Niyazi Mar 5 '11 at 21:00

Yes, as per Ryan's comment you can just take an infinite direct sum of finite groups. However the more interesting problem is: are there (infinite) $\textit{finitely generated}$ groups with all elements of finite order?

The answer to this was open for a long time, but it is indeed yes. In fact this was known as Burnside's problem

The first examples were given by Golod & Shafarevich.