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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 alot a lot of info on the wikipedia page |
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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 $\textit{finitely generated}$ 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 alot of info on the wikipedia page |
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