The center of $\langle x,y:x^3=y^2=1\rangle$

Question

Let $G$ be the group given by the following presentation $$ G=\langle x,y:x^3=y^2=1\rangle $$ I want to find $Z(G)$. I notice that $\langle x\rangle\cap\langle y\rangle=\{1\}$, Does that means that $Z(G)=\{1\}$?

Answer 1

There is no reason why the intersection being trivial would imply a trivial center. In the dihedral group of order $8$, the group is generated by a rotation and a reflection, the cyclic subgroups they generate have trivial intersection, yet the center is not trivial.

The group $G$ is isomorphic to $C_3*C_2$, the free product of a cyclic group of order $3$ and one of order $2$. Free products of nontrivial groups are centerless, so $G$ is centerless.