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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