Let $|a| = 4$ and $|b| = 2$. Write $\left \langle a \right \rangle \times \left \langle b \right \rangle$ as the internal direct product of two proper subgroups in every possible way.
Let $G=\left \langle a \right \rangle \times \left \langle b \right \rangle$, we have that $|G|=8$. I've been looking for a way to write $G$ as the internal product of proper subgroups and so far I can think of two ways: $G=\left \langle a \right \rangle \left \langle b \right \rangle$, since $\left \langle a \right \rangle \times \left \langle b \right \rangle \cong \left \langle a \right \rangle \left \langle b \right \rangle$ and $G=\left \langle a^3 \right \rangle \left \langle b \right \rangle$. Is this correct? are there more possibilities?