TITLE: What is a short exact sequence telling me?
QUESTION [34 upvotes]: Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a sort of curiosity, what is it really telling me?

REPLY [4 votes]: "The kernel of $B \to C$" is often not a satisfactory description of a group. Having a group $A$ that it is isomorphic to (along with a given isomorphism) is very useful.
"The quotient $B/A$" is often not a satisfactory description of a gruop. Having a group $C$ that it is isomorphic to (along with a given isomorphism) is very useful.