An exact sequence is *short* if it has exactly 3 elements:

1 → A → B → C → 1

This is the most important (

finite) exact sequence, and the shortest one to demonstrate any interesting

structure.

We have (identifying "1" as a subset of each of A,B,C) 1=Im(1→A)=Ker(A→B), so the mapping A→B is a monomorphism (this is true for the first 2 arrows in *any* exact sequence). On the other side, C=Ker(C→1)=Im(B→C), so the mapping B→C is an epimorphism (this is true for the last 2 arrows in any exact sequence).

Finally, we have Im(A→B)=Ker(B→C). Think of A→B as describing an inclusion of A in B (it is one to one) -- if B is a group, think of A as its subgroup. Think of B→C as describing C as a quotient of B (if we're dealing in groups, C is (isomorphic to) a quotient group of B). By the first isomorphism theorem, C=Im(B→C) is isomorphic to B/Ker(B→C) = B/A.

So a short exact sequence is a precise description of the relationship between a (normal, for groups) subobject and the quotient it generates.

For groups, this structure can be very complex. Short exact sequences are *hard*, because they describe a hard concept.