TITLE: What is the deal with the three isomorphism theorems?
QUESTION [10 upvotes]: I've been having lectures in group theory with Hungerford's book. We were presented with the following theorem:



And then with:





Previous to the lectures about them, I was understanding most of the stuff, that is: I kinda could figure out the motivation for things. But when I came into these theorems, things got pretty weird for me. I don't understand the motivation for them. I have the following guesses:

Theorem 5.6 is used to prove Corollary 5.7 but I think it has a utility on its own: If given an homomorphism $f: G \to H$ with $N\lhd G$, we have a unique homomorphism $\overline{f} : G/N\to H$, then we could use this to know what homomorphisms could exist betwen $G$ and $H$ and this depends only on the normal subgroups of $G$. Does that make sense?

For the isomorphism theorems, I noticed they are corollaries so they must be very important. One of the reasons I conjectured for them is that we can construct very interesting isomorphisms. But why are those isomorphisms interesting? It's not clear to me what interesting stuff one could do with them.


I know my guesses are maybe too obvious or wrong, but I can't figure out on my own the answer to that question. Can you help me?

REPLY [15 votes]: Let’s say you have a group you are interested in. One way to try to understand the group is to stare at it intently until you see interesting things about it.
But the big insight of abstract algebra in general (and of Category Theory) is that you can also learn a lot about the group by considering morphisms, both into and out of the group. In fact, the Yoneda embedding tells you that understanding all the ways that all groups can map into $G$ will completely determine the isomorphism type of $G$, and that understanding all the ways in which you can map from $G$ to all other groups will completely determine the isomorphism type of $G$ as well.
In addition, one can view these maps as casting “shadows” of the object $G$; with some luck, the different shadows will each be simpler than the group, and perhaps we can piece together enough “shadows” to get a full picture of what the group looks like without having to look at the full group (in turns out that is not always possible, as witnessed by simple groups, but it works for many other types of groups).
The First Isomorphism Theorem tell you that if you want to understand the way that $G$ maps into other groups, you don’t need to go scour the universe to find those groups: you can “realize” any map $f\colon G\to H$ by finding a suitable normal subgroup $K$ of $G$, and then the image of $G$ under $f$ will behave exactly the same way as the image of $G$ under the canonical (and usually well-understood) quotient map $\pi\colon G\to G/K$ behaves. So in a sense you can understand all those images of $G$ by staring at suitable shadows of $G$: just the quotient ones.
The Third Isomorphism Theorem tells you that you that iterating the process won’t get you any new information: if you take a quotient of $G/K$, you are still just taking a quotient of $G$.
The Second Isomorphism Theorem tells you how the subgroups of $G$ behave under the map $G\to G/N$: namely the image of $K$ in the map will be $K/(N\cap K)$, but this will be isomorphic to $KN/N$. When you connect it with the Fourth (or Lattice) isomorphism theorem, which you did not quote, it provides complete information about what happens to the subgroups of $G$ under the map $G\to G/N$: the subgroups that contain $N$ are mapped to the subgroups of $G/N$ in a one-to-one, inclusion preserving, normality preserving manner, while the subgroups of $G$ that do not contain $N$ correspond, under the Second Isomorphism Theorem, to certain subgroups that do contain $N$.
These theorems are valid in a much more general setting: they apply, suitably modified to account for some technical issues, to any kind of algebraic structure: semigroups, monoids, rings, lattices, and more. They form a backbone of investigations, tools used over and over again, because one is always looking at homomorphisms to better understand the structures, and because one is always looking at quotients to simplify matters and try to get a better sense of the object one is studying.