How to approach proofs similar to "Show a group, $G$, is infinite if $G = \langle r, s, t\mid rst = 1\rangle $"
I have not worked much with relations and tend to get lost in notation. I am practicing solving problems like the one in the title but am having a hard time as I am not sure the tricks to try or areas to investigate first in trying to make a proof. What are some hints for starting a proof about some quality of a group defined by a relation?
So far the only relations I know about are the dihedral groups of order $2n$, the quaternions, and cyclically generated groups so comparisons to how we show properties of those might be illuminating.
Best Answer
$G$ is the set of words on $r,s,t$ subject to the relation $rst=1$.
The relation $rst=1$ means that you can replace every occurrence of $t$ by $(rs)^{-1}=s^{-1}r^{-1}$.
Therefore, $G$ is the set of words on $r,s$, that is, the free group in two letters.
Alternatively, the set $\{1,r,r^2, r^3, \dots \}$ is an infinite subset of $G$ because these words do not contain $s$ or $t$ and so cannot be further reduced or to one another.
(By words on $S$, I mean words on the elements of $S$ and their inverses.)