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. 



Properties of a special finitely presented groups
Your group $G$ is *not* solvable since it has a quotient isomorphic
to ${\rm S}\_5$. You can see this with [GAP](http://www.gap-system.org) 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.


