关于用商群替换正合序列项的可行性问询

Hey there! Let's break down your question and clear up the confusion step by step.

First, the core issue with your proposed short exact sequence #

Your idea to build 0 → B/A → D/C → F/E → 0 runs into a fundamental problem right away: the quotient groups D/C and F/E are not well-defined in general.

For a group quotient G/H to make sense, H must be a normal subgroup of G—and crucially, H has to be a subset of G. In your original long exact sequence:

• C is a separate group from D, connected only by a homomorphism C→D. Unless this homomorphism is injective (which only happens if Im(B→C) = 0, a special case), C is not a subgroup of D, so D/C isn't a valid quotient group.
• The same goes for F/E: E is not a subgroup of F (since E→F is surjective, not injective, in the exact sequence), so F/E also has no meaning here.

Your correct observation about the kernel/image relationship #

You were right to note that Ker(B→C) = Im(A→B)—that's exactly what the exactness of 0→A→B→C tells us! By the First Isomorphism Theorem, this means B/A ≅ Im(B→C) (since A ≅ Im(A→B) via the injective map A→B). That part of your reasoning checks out.

What you can do instead #

If we restrict to Abelian groups (where all subgroups are normal, making quotients easier to work with), we can use cokernels (defined as Coker(f) = Y/Im(f) for a homomorphism f:X→Y) to construct meaningful sequences from the long exact sequence, but not the one you proposed.

For example, from your original sequence:

• We get a short exact sequence 0→A→B→Im(B→C)→0 (from the first three terms)
• Another short exact sequence 0→Im(B→C)→C→Im(C→D)→0 (using the exactness at C)
• And so on, working through each "exactness point" in the long sequence to extract smaller short exact sequences.

Alternatively, tools like the Snake Lemma can help build new exact sequences from commutative diagrams involving your long exact sequence, but that's a different direction than your initial idea.

Why your approach might feel intuitive (and where it goes wrong) #

You might be mixing up group exact sequences with module exact sequences (modules are Abelian groups with extra structure). Even in modules, though, D/C wouldn't be valid—you'd need D/Im(C→D) instead, and there's no natural way to link that to B/A and F (which is isomorphic to E/Im(D→E)) into a short exact sequence without additional constraints.

内容的提问来源于stack exchange，提问作者H. Hughes