There is a lemma that I'm trying to understand in my algebra class and I can't get it done. It says:
Given $G$ a group, let $A, B, H \leq G$ be subgroups of $G$ such that $A \triangleleft B$ and $H \triangleleft G$. Then, $HA \triangleleft HB$ and $\frac{HB}{HA}$ is isomorphic to some quotient of $B/A$.
The second part is pretty ok to me and I got the morphism right away, but as much as it seems stupid, I can't get the normality done, and I've tried a lot. And it is not only this problem, I just seem to be HORRIBLE proving normality of stuff. Any tips on that?
For any $a\in A$ and $b\in B$, we have $$bab^{-1}=a_1$$ for some $a_1\in A$. Note that $a_1H=Ha_1$ and $bH=Hb$. Then $$(Hb)Ha(b^{-1}H)=HbH(ab^{-1})H=HbHb^{-1}a_1H=Ha_1.$$