If $H$ and $K$ are abelian subgroups of a group $G$, then $H\cap K$ is a normal subgroup of $\left\langle H\cup K\right\rangle$.
I proved $H\cap K$ is a subgroup and need to prove it is normal subgroup of $\left\langle H\cup K\right\rangle$. But isn't it obvious that $H\cap K$ is normal? All the elements in $H\cap K$ communicates with all the elements in $H$ and $K$, hence $H\cap K$ is normal. Am I missing something?
Best Answer
Yes, every element of $H\cap K$ commutes with every element of $H$ and with every element of $K$. In my opinion, you should add a proof of that fact that it follows from this that every element of $H\cap K$ commutes with every element of $\langle H\cup K\rangle$. You can say, for instance, that every element of $\langle H\cup K\rangle$ can be written as a product of elements of $H\cup K$ and then use that fact.