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Recently, when I was working with Cayley graphs, I faced up with a special group. The original group is as follows: $$G:=<a,b,c|ab=ba,a^{10}=cbc^{-1}>.$$

We can show that this group can be rewrite as follows: $$G=<x,y|xy^{-1}x^{10}yx^{-1}y^{-1}x^{-10}y=1>.$$

In general, there is not any special things about the power $10$ of the element $a$ and for any integer $n$, we can transform the original group to the second form, where the power $10$ of $x$ will be replace with $n$.

Are there any special things about this group? For example, is this group solvable? Can we prove that this group has a torsion element or it is a torsion-free group?

Thanks for your comments and answers.

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    $\begingroup$ Read about HNN extensions. $\endgroup$
    – Uri Bader
    Commented Apr 17, 2016 at 8:35
  • $\begingroup$ Thanks for the introducing HNN extensions. It seems interesting, as I was not familiar with this topic. $\endgroup$
    – Shahrooz
    Commented Apr 17, 2016 at 9:41
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    $\begingroup$ It is certainly torsion-free. $\endgroup$
    – Derek Holt
    Commented Apr 17, 2016 at 11:38
  • $\begingroup$ Dear Holt, I can not see why it is torsion free? Would you please say some more details? $\endgroup$
    – Shahrooz
    Commented Apr 17, 2016 at 17:32
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    $\begingroup$ An HNN extension is torsion-free if and only if its base group is torsion free. As user89334 said, you need to learn about HNN extensions in order to understand this group. $\endgroup$
    – Derek Holt
    Commented Apr 17, 2016 at 20:52

1 Answer 1

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Your group $G$ is not solvable since it has a quotient isomorphic to ${\rm S}_5$. You can see this with GAP as follows:

gap> F := FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> AssignGeneratorVariables(F);        
#I  Assigned the global variables [ a, b, c ]
gap> G := F/[Comm(a,b),a^10/(c*b*c^-1)];
<fp group of size infinity on the generators [ a, b, c ]>
gap> low := LowIndexSubgroupsFpGroup(G,5);;
gap> H := First(low,H->Index(G,H)=5
>                      and Size(Action(G,RightCosets(G,H),OnRight)) >= 60);
Group(<fp, no generators known>)
gap> Q := Action(G,RightCosets(G,H),OnRight);
Group([ (4,5), (), (1,2,3,4) ])
gap> StructureDescription(Q);
"S5"

You can rewrite the second presentation of your group $G$ as $$ G \ = \ \langle x, y \ | \ [x,(x^{10})^y] = 1 \rangle. $$ From this you see quickly e.g. that $G$ is torsion-free, as Derek Holt has already written in his comment without explanation.

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  • $\begingroup$ Thanks dear Kohl, is there any method such that we can check in GAP that the group $G$ is orderable? $\endgroup$
    – Shahrooz
    Commented Apr 17, 2016 at 9:37
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    $\begingroup$ All one-relator groups are left-orderable. $\endgroup$
    – HJRW
    Commented Apr 17, 2016 at 13:01
  • $\begingroup$ Dear HJRW, I saw it after some search in google. But, would you please introduce me some good reference about the proof of this fact and some related similar theorems? $\endgroup$
    – Shahrooz
    Commented Apr 17, 2016 at 17:34
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    $\begingroup$ It's a standard fact (I should of course have said all torsion-free one-relator groups are left-orderable). The only proof I know is to use the ideas of arXiv:1410.2540 to prove that they're locally indicable, and then to quote the well known theorem that locally indicable groups are left-orderable. $\endgroup$
    – HJRW
    Commented Apr 17, 2016 at 20:20

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