Can a subgroup generated by two finite subgroups be infinite? Let $H$ and $K$ be two subgroups of $G$ of finite order. Can $HK$ be infinite?

Ok, since it seems that the general consensus is to look at the subgroup generated by $HK$, here is another example.
The modular group $SL_2(\mathbb Z)$ is generated by two matrices of finite order:
$$\begin{pmatrix} 0&1\\-1&0\end{pmatrix},\text{ and }\begin{pmatrix} 0&1\\-1&0\end{pmatrix}\begin{pmatrix} 1&1\\0&1\end{pmatrix}=\begin{pmatrix} 0&1\\-1&-1\end{pmatrix};$$
the former has order $4$ and the latter, order $3$. And of course, $SL_2(\mathbb Z)$ is infinite.
