Exercise 18, § 5

Let $(m, n)$ be a pair of natural integers such that $1< m < n$. Let G be the group defined by the presentation $\langle x, y;x^m=y^n\rangle$. Let $z=x^m$ and let C be the subgroup of G generated by $z$.

a) Prove that C is contained in the center of G.

b) Prove that $G/C$ is isomorphic to the free product $(\mathbf{Z}/m\mathbf{Z})*(\mathbf{Z}/n\mathbf{Z})$.

c) Deduce that C is the center of G.

d) Let $(p, q)$ be a pair of natural integers such that $1< p < q$. Suppose that the group $G'$ defined by the presentation $\langle x, y;x^p=y^q\}$ is isomorphic to G. Prove that $(p, q) = (m, n)$.

e) Prove that the quotient of G by its derived group is isomorphic to $\mathbf{Z}$.