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? 


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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 

