I've understood that there are finitely generated groups which are also locally finite groups (an infinite finitely generated group which has no subgroups of finite index that are no trivial), but I can't find an example so such group.

I would love if someone could point me to such an example

  • $\begingroup$ Perhaps I misunderstood something, but a locally finite group (=LFG) is one in which any f.g. subgroup is finite, so if the LFG is itself finitely generated then it is finite...am I missing something? Your definition between parentheses seems to add to the LFG the condition of being infinite and f.g., but why would that be so? $\endgroup$ – DonAntonio May 27 '16 at 9:57
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    $\begingroup$ He doesn't mean locally finite. There are infinite finitely generated (even finitely presented ) simple groups. See for example en.wikipedia.org/wiki/Higman_group $\endgroup$ – Derek Holt May 27 '16 at 10:12
  • $\begingroup$ @DerekHolt Thank you, now I see what he most probably meant $\endgroup$ – DonAntonio May 27 '16 at 10:27
  • $\begingroup$ @tzoorp You seem to have a non-standard definition of locally finite. But you are looking for an example of an infinite finitely-generated group with no non-trivial subgroups of finite index. Is that correct? $\endgroup$ – almagest May 27 '16 at 11:37
  • $\begingroup$ @almagest yes, that is what I'm looking for.. so what is the standard definition for locally finite group? just out of curiosity $\endgroup$ – tzoorp May 27 '16 at 11:51

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