# Properties of a special finitely presented groups

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 the introducing HNN extensions. It seems interesting, as I was not familiar with this topic. Commented Apr 17, 2016 at 9:41
• It is certainly torsion-free. Commented Apr 17, 2016 at 11:38
• Dear Holt, I can not see why it is torsion free? Would you please say some more details? Commented Apr 17, 2016 at 17:32
• 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. Commented Apr 17, 2016 at 20:52

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.

• Thanks dear Kohl, is there any method such that we can check in GAP that the group $G$ is orderable? Commented Apr 17, 2016 at 9:37
• All one-relator groups are left-orderable.
– HJRW
Commented Apr 17, 2016 at 13:01
• 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? Commented Apr 17, 2016 at 17:34
• 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.
– HJRW
Commented Apr 17, 2016 at 20:20