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 (edit: but finitely generated) group G such that for all g in G, g is finite? If so, how many such groups exist? 

closed as too localized by Ryan Budney, J.C. Ottem, Ian Agol, Felipe Voloch, Pete L. Clark Mar 5 '11 at 21:12This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


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 

