Isomorphism of Galois groups of the intersection

Question

Let $E/F$ and $E/F'$ and $E/F\cap F'$ be separable and normal field extensions. I am trying to show that

$\text{Gal}(E/F\cap F') \simeq \langle \text{Gal}(E/F) \cup \text{Gal}(E/F')\rangle$

Answer 1

The groups on the right hand side are subgroups of $Gal(E/F\cap F')$ . In general, the left hand side will be the generated by these subgroups. However, that may not equal the product the way it is written. It works OK if at least one of the extensions $F/F\cap F'$, $F'/F\cap F'$ is normal