Galois Connections
Let's be honest, the correspondence between Galois groups and field extension is pretty hott. The first time I saw this I was duly impressed. However, about two years ago, I learned about universal covering spaces. Wow! I swear my understanding of covering spaces doubled when the prof told me that this was a "Galois correspondence for fundamental groups and covering spaces".
Again here is a link!
I find that, for elementary group theory problems, one of the best places to start is by asking yourself if (how) groups actions could be used to express this. Then you have some very powerful (while still elementary) theorems that you can use.
For your examples, I point you to User-33433's answer.
For another example which was found in a homework problem in the abstract algebra class that I just finished:
Show that any group (including infinite) that contains a proper subgroup of finite index also contains a proper normal subgroup of finite index.
One way to prove it is to go through all of the motions of proving that the intersection of finite index subsets is, again, finite index, extending this by induction, looking at $N=\bigcap_{g\in G}gHg^{-1}$, proving that this normal and is actually a finite intersection by the Orbit-Stabilizer Theorem when $G$ acts on the cosets of $H$ by conjugation, and finally using the result on finite intersections of finite index sets that you would have proved.
The easier, and more powerful, way to show this result (and even more, as we'll see) is the following which comes from considering the action of $G$ on the cosets of $H$ by conjugation as the important feature, rather than just a means to show that the previous set $N$ is a finite intersection.
Let $H\leq G$ be of finite index $n$. Then, $G$ acts on the cosets of $H$ by conjugation, and this induces a map $\varphi:G\to\text{Sym}\left(G/H\right)\cong S_{n}$. The kernel of this map is normal, and $G/\ker\varphi$ isomorphic to a subgroup of $S_{n}$, so $|G:\ker\varphi|\leq n!$, as required.
This proves that, not only do we have a proper normal subgroup of finite index, but we have one of index $\leq n!$. So, group actions are very powerful and allow you to go straight to the core of many elementary group theory problems, rather than having you flounder around for a bunch of other results just to scratch the surface.
Best Answer
The connection between the two is explained in an old blog post of mine. The Wikipedia article has more details.