Is there an infinite group whose elements all have finite order? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-26T08:56:09Z http://mathoverflow.net/feeds/question/57493 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57493/is-there-an-infinite-group-whose-elements-all-have-finite-order Is there an infinite group whose elements all have finite order? wjomlex 2011-03-05T20:09:12Z 2011-03-05T20:57:55Z <p>Is there an infinite (edit: but finitely generated) group G such that for all g in G, |g| is finite?</p> <p>If so, how many such groups exist?</p> http://mathoverflow.net/questions/57493/is-there-an-infinite-group-whose-elements-all-have-finite-order/57496#57496 Answer by Owen Sizemore for Is there an infinite group whose elements all have finite order? Owen Sizemore 2011-03-05T20:16:29Z 2011-03-05T20:48:32Z <p>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? </p> <p>The answer to this was open for a long time, but it is indeed yes. In fact this was known as Burnside's problem</p> <p>The first examples were given by Golod &amp; Shafarevich.</p> <p>There is a lot of info on the wikipedia page</p> <p><a href="http://en.wikipedia.org/wiki/Burnside%27s_problem" rel="nofollow">http://en.wikipedia.org/wiki/Burnside%27s_problem</a> </p>