11

I have the following group presentation:

$G=\left\langle a,b,c\ |\ a^2,b^{11},c^2,(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},(ac)^3,(bc)^2\right\rangle$

Is $G$ finite?

GAP's Size(G) runs out of memory pretty quickly. No surprise there. I also tried using the ideas in the answer here to look for homomorphisms onto certain small simple groups, with no luck. And I used LowIndexSubgroups() to look for subgroups up to index $30$ with no results.

It's also worth noting that:

  • $a$ and $b$ generate a subgroup isomorphic to $M_{11}$ (The Mathieu Group on $11$ points)

  • $a$ and $c$ generate a subgroup isomorphic to $D_6$ (The Dihedral group of size $6$)

  • $b$ and $c$ generate a subgroup isomorphic to $D_{22}$ (The Dihedral group of size $22$)

Olexandr Konovalov
  • 7,186
  • 2
  • 35
  • 73
Josh B.
  • 3,476
  • 2
    Just out of curiosity, where did this presentation come from? – lulu Feb 14 '18 at 02:00
  • @lulu I've been exploring some finite groups presented with two generators, then tacking on $c$ and the relations involving $c$. Sometimes you do get a finite group. – Josh B. Feb 14 '18 at 02:03
  • Let us continue this discussion in chat. – verret Feb 14 '18 at 03:56
  • 2
    Yes, if you leave out one of the letters, the other two will present a group as listed in the question. But if you combine such presentations, it is not guaranteed -- see the example here -- that such groups survive as subgroup. A more stupid example is to take a free cyclic group generated by $a$ and then add relations $b^2$, $abab$ and $ba^2b$. Suddenly the infinite subgroup $$ is only cyclic of order $2$. – ahulpke Feb 14 '18 at 04:01
  • Thank you. That is a good point. It is possible the whole thing could collapse and be the trivial group. – Josh B. Feb 14 '18 at 04:03
  • 10
    I checked that it has no proper subgroups up to index $40$, and no simple images up order up to $500$ million. I don't have any further ideas of things to try. The general problem of deciding whether a group defined by a finite presentation is trivial, finite or infinite, etc is known to be undecidable, so it is unsurprising that presentations arise from time to time which are difficult to resolve. – Derek Holt Feb 14 '18 at 09:44
  • @DerekHolt Thanks for doing those calculations. At least I didn't miss an obvious trick. I'm not quite done with this question yet though! – Josh B. Feb 15 '18 at 00:34

0 Answers0