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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    $\begingroup$ The direct sum of infinitely many copies of $\mathbb Z_2$. $\endgroup$ Mar 5, 2011 at 20:12
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    $\begingroup$ Edit to specify "infinite finitely generated group"? Anyone? $\endgroup$ Mar 5, 2011 at 20:24
  • $\begingroup$ Yes, Grigorchuk group is a finitely generated infinite 2-group. I.e, for each $g\in G$ there is an $n$ such that g^(2^n)=1. $\endgroup$
    – Niyazi
    Mar 5, 2011 at 21:00

1 Answer 1


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


  • $\begingroup$ Nice answer, +1. Nitpicky requests: will you edit to say you want your groups to be infinite cardinality, and remove the "alot"? (hyperboleandahalf.blogspot.com/2010/04/…) $\endgroup$ Mar 5, 2011 at 20:23
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    $\begingroup$ Done. In general, though, let's not worry about such picayune errors. I recommend a good dose of Language Log to those who worry alot about correct grammer and spelling. ;-) $\endgroup$
    – Todd Trimble
    Mar 5, 2011 at 20:56
  • $\begingroup$ Heh, thanks. I care a lot more about the precision in math than grammar on this forum, but I couldn't resist the addition; my genuine-linguist friends have as yet failed to convince me that prescriptivism is always bad. $\endgroup$ Mar 5, 2011 at 21:02
  • $\begingroup$ Elizabeth, even so, there's a huge difference between a grammatical error and a typographical error. $\endgroup$
    – HJRW
    Mar 6, 2011 at 6:27

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