Let $G$ be the group generated by $a,b,c$ with relations $ab=b^2a$, $bc=c^2b$, and $ca=a^2c$. Show that $G$ is trivial.
A related problem is that for $G_1$ generated by $a,b,c,d$ with similar relations is NOT trivial.
I'm not sure how to start. I know that word problem for groups does not have algorithm to solve. But we should be able to tackle this particular case.
Elementary method is preferred. I am not familiar with advanced methods for word problems.
