# An example of a simple infinite 2-group

Is there an example of a simple infinite $2$-group?
• 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.
• 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. – YCor Jan 16 '15 at 16:01