6
$\begingroup$

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.

$\endgroup$
3
  • 11
    $\begingroup$ 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
    Jan 16, 2015 at 16:01
  • 1
    $\begingroup$ @YCor: Why don't you write an answer, instead of putting your answer in a comment? -- This question certainly deserves it! $\endgroup$
    – Stefan Kohl
    Jan 17, 2015 at 14:30
  • $\begingroup$ OK, I answered the question on StackExchange. $\endgroup$
    – YCor
    Jan 17, 2015 at 15:35

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.