normal subgroup generated by a subgroup Let $G= \langle g_1,g_2 \rangle$, and let $H\leq G$ given by $H=\langle g_1 g_2 g_{1}^{-1}g_{2}^{-1}\rangle$. 
What is the normal subgroup of $G$ generated by $H$? 

Let $N$ be the normal closure of $H$. Since $[g_1,g_2] \in N$, $[g_1N,g_2N]=1N$ and $G/N$ is generated by two commuting elements, so is commutative as a whole. Hence $G' \leq N$. However, $G'$ is a normal subgroup containing the commutator $[g_1,g_2]$, so $H \leq G'$ and so $N \leq G'$. Hence $N=G'$.
