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

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  • 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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  • $\begingroup$ I strongly suspect that it is not known whether such an example exists. You could try asking on mathoverflow. $\endgroup$ – Derek Holt Jan 16 '15 at 15:42
  • $\begingroup$ I'll be happy to know that it is not known. :-) I'll try ask on mathoverflow also. $\endgroup$ – W4cc0 Jan 16 '15 at 15:48
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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.

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