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?



Is there an infinite group whose elements all have finite order?
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.


There is a lot of info on the wikipedia page


<http://en.wikipedia.org/wiki/Burnside%27s_problem> 


