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I've asked this question before on Mathematics, and they suggested me to ask here (Link).

Is there an example of a simple infinite $2$-group?

Informations

  • If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian.

  • Take the subgroup generated by the elements of order $2$, it must coincide with $G$. If we have a periodic subgroup generated by two elements of order $2$, like $\langle a,\, b\rangle$, it must be finite.

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    $\begingroup$ Yes. Take the Burnside group on 2 generators and exponent $2^k$ for large $k$, which is known to be infinite (it's hard!). By the restricted Burnside problem (it's hard too), it has a minimal finite index subgroup, say $H$; hence $H$ is infinite, finitely generated and has no nontrivial finite quotient. Hence $H$ admits a simple quotient, which is necessarily infinite. $\endgroup$
    – YCor
    Commented Jan 16, 2015 at 16:01
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    $\begingroup$ @YCor: Why don't you write an answer, instead of putting your answer in a comment? -- This question certainly deserves it! $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 17, 2015 at 14:30
  • $\begingroup$ OK, I answered the question on StackExchange. $\endgroup$
    – YCor
    Commented Jan 17, 2015 at 15:35

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