Let $G=\langle a,b|a^2=b^3=1,(ab)^n=(ab^{-1}ab)^k \rangle$. Prove that $G$ can be generated with $ab$ and $ab^{-1}ab$. And from there, $\langle(ab)^n\rangle\subset Z(G)$.
Problem wants $H=\langle ab,ab^{-1}ab \rangle$ to be $G$. Clearly, $H\leqslant G$ and after doing some handy calculation which takes time I've got:
$ab^{-1}=(ab^{-1}ab)(ab)^{-1}\in H$
$b=b^{-2}=(ab)^{-1}ab^{-1}\in H$
$a=(ab)b^{-1}\in H$
So $G\leqslant H$ and therefore $G=H=\langle ab,ab^{-1}ab\rangle$.
For the second part, I should prove that $N=\langle(ab)^n\rangle\leqslant Z(G)$.
Please help me.
Thanks.