Wreath Product of Two Finitely Generated Groups Let $G$ and $H$ be two finitely generated groups, and let $W = G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated. In class today, we were showed this and told that it was obvious. However, I do not see how it is obvious. How is this obvious?
 A: Consider the set of elements of the form $(a,e_H)$ where
$$\begin{cases}
a_h \in \langle G \rangle & h = e_H \\
a_h = e_G & \text{otherwise}
\end{cases}$$
together with the set of elements of the form $\{e_G\}^H \times \langle H \rangle$. More concisely,
$$\langle G \wr H \rangle = \left(\left(\{e_G\}^{H-\{e_H\}} \times \langle G \rangle^{\{e_H\}}\right) \times \{e_H\}\right) \cup \left(\{e_G\}^H \times \langle H \rangle\right)$$
Then, assuming $e_G \not\in \langle G \rangle \lor e_H \not\in \langle H \rangle$,
\begin{align}
|\langle G \wr H \rangle|
&= |\left(\left(\{e_G\}^{H-\{e_H\}} \times \langle G \rangle^{\{e_H\}}\right) \times \{e_H\}\right) \cup \left(\{e_G\}^H \times \langle H \rangle\right)| \\
&= |\left(\{e_G\}^{H-\{e_H\}} \times \langle G \rangle^{\{e_H\}}\right) \times \{e_H\}| + |\{e_G\}^H \times \langle H \rangle| \\
&= |\{e_G\}^{H-\{e_H\}}| |\langle G \rangle^{\{e_H\}}| |\{e_H\}| + |\{e_G\}^H| |\langle H \rangle| \\
&= |\{e_G\}|^{|H - \{e_H\}|} |\langle G \rangle|^{|\{e_H\}|} 1 + |\{e_G\}|^{|H|} |\langle H \rangle| \\
&= 1^{|H-\{e_H\}|} |\langle G \rangle|^1 + 1^{|H|} |\langle H \rangle| \\
&= 1 |\langle G \rangle| + 1 |\langle H \rangle| \\
&= |\langle G \rangle| + |\langle H \rangle|
\end{align}
Since $\langle G \rangle$ and $\langle H \rangle$ are finite, $\langle G \wr H \rangle$ is also finite.
