Often I see things like this:
$$0\rightarrow G \rightarrow H \rightarrow \mathbb{Z}_2 \rightarrow 0$$
Which has something to do with Lie groups or groups in general. But I have never been able to work out exactly what they mean.
Is there some explanation of this and why exactly it is important?
I understand about groups, lie groups, simple groups and normal groups etc.
Often if one has a family of groups, and important maps between them, it is nice to lay them out in a diagram. In this case, we write each group separately, and draw (possibly labelled) arrows connecting them to denote the maps.
Two important families of diagrams are:
$G_1 \to G_2 \to G_3 \to G_4 \to \cdots$
and
$\cdots H_4 \to H_3 \to H_2 \to H_1$
These are important because they often have limits, which are groups $\varinjlim G_i$ and $\varprojlim H_i$ (respectively) such that for every $j$, there is a map
One can also impose certain restrictions on how the maps in the diagram compose, and the most important such restriction is called exactness. If we have objects $G_i$ and maps $f_i : G_i \to G_{i+1}$, then the sequence is called exact when
$Ker(f_{i+1}) = Im(f_i)$
This says that everything $f_i$ hits is immediately killed by $f_{i+1}$, and moreover $f_{i+1}$ kills of exactly what $f_{i+1}$ sees - no more, no less.
A typical example of an exact sequence is called Short Exact Sequence and looks like this:
$1 \to A \xrightarrow{\alpha} B \xrightarrow{\beta} C \to 1$
(here $1$ denotes the trivial group)
Now, by exactness, we know that $\alpha$ is injective. There is only one group homomorphism $1 \to A$, namely the map sending the identity to the identity. But by exactness, the image of that map (which is trivial) must be the kernel of $\alpha$! So $Ker(\alpha)$ is trivial and $\alpha$ is injective.
Similarly, we know that $\beta$ is surjective. There is only one group homomorphism $C \to 1$, and it sends every element to the identity of $1$. That is, every element of $C$ is in the kernel of this map! Then, by exactness, the image of $\beta$ must be all of $C$.
Finally, we use exactness one last time to see that $Im(\alpha) = Ker(\beta)$. But then, by the first isomorphism theorem (since $\beta$ is surjective), $B / Im(\alpha) \cong C$. And since $\alpha$ is injective, we have (also by the first isomorphism theorem) that $A \cong Im(\alpha)$. So the exact sequence is telling us that $B / A \cong C$.
Ok, so this is nice and all, but it doesn't tell us anything we didn't already know. The limits above can just as easily be written without these diagrams, and the exact sequences tell us nothing that the first isomorphism theorem didn't. So why bother?
The reason to use this notation is not because it tells us new things, but because it gives us a better way to structure our thoughts. This notation is extremely clear, and once you get used to reading it, it can say with one picture what would take a paragraph to write down without it.
Additionally, as one starts dipping their toes into category theory, writing down things that we know using these diagrams can pay large dividends. Category Theory, like these diagrams, only tells us things we already knew. But by giving us a new, more organized, framework to think in, we can see patterns that we hadn't before.
As a "practical" application of these tools, let's work on a problem which is obviously of interest to many people. A general principle in math is to try to take a complicated object, break it into smaller pieces, and analyze those separately (or inductively) and then glue the pieces back together to recover information about the original object of interest.
If we want to apply this framework to groups, we need to understand how to glue two groups together to get a new group. Now, just as with numbers we can glue $a$ and $b$ together in any number of ways ($a+b$, $a \times b$, $a^b$, etc.), there are a variety of ways to glue two groups $K$ and $Q$ together in order to get a new group $G$.
The problem, then, is understanding (for a fixed $K$ and $Q$) which groups $G$ fit into the following short exact sequence:
$1 \to K \to G \to Q \to 1$
For the rest of the post, let's work with abelian groups, although a variant on this approach will work for all groups.
It turns out that these extensions are governed by the "Cohomology Groups" $H^n(Q;K)$, and indeed $H^2(Q;K)$ can be identified with exactly those $G$ "extending $Q$ by $K$". That is, those groups making $1 \to K \to G \to Q \to 1$ exact.
Ok, how might one calculate $H^2(Q;K)$, then? Well, it turns out to be yet another diagram. Without getting lost in the details, we write down an exact sequence (of G-modules) with some bonus properties (it is projective)
$\cdots \to M_3 \to M_2 \to M_1 \to \mathbb{Z} \to 1$
Then we hit each element of this sequence with $Hom(-,K)$. That is, we consider the abelian group $Hom(M_i,K)$ of homomorphisms from $M_i$ to $K$ (with addition defined pointwise) to get a new sequence
$\cdots \to Hom(M_3,K) \to Hom(M_2,K) \to Hom(M_1,K) \to Hom(\mathbb{Z},K) \to 1$
Of course, this new sequence might not be exact anymore, why would it be? The magical thing is that $H^n(G,K)$, which we want to understand to solve our problem, measures how inexact this new sequence is.
So now we see that the notion of exactness is important in order to study the (very practical) notion of a group extension. Of course, we cannot properly formulate the notion of exactness without first writing our groups and homomorphisms in these diagrams. I emphasize that these diagrams don't actually do any math, but they make it easier for us to think about the problems at hand. Modern mathematics is steeped in this categorical language, and for good reason - it's really helpful. And diagrams and short exact sequences are the first step on the road to gaining comfort with these categorical methods.
Hope this helped ^_^
