Normal subgroup of a finite group

Question

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?

Answer 1

Yes (and finiteness of $G$ is not needed). First, note that if $x_1 \in X_1$, $x_2 \in X_2$, then $x_2x_1 \in X_1X_2$. Indeed, if $X_1$ is normal, then $x_2x_1 = (x_2x_1x_2^{-1})x_2 \in X_1X_2$, and if $X_2$ is normal, then $x_2x_1 = x_1(x_1^{-1}x_2x_1) \in X_1X_2$.

Thus $X_1X_2$ is in fact a subgroup, and $X_1X_2 = X_2X_1$. Then any element of $\langle X_1, X_2 \rangle$ is contained in a finite product $X_{i_1}X_{i_2}\ldots X_{i_k}$, where $i_j \in \{1,2\}$, but this is contained in $X_1X_2$.

The same holds if we only assume $X_1 \subseteq N_G(X_2)$, or $X_2 \subseteq N_G(X_1)$.