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Given a short exact sequence of finite abelian groups, is it possible to classify what groups can show up in the middle based on the kernel and the cokernel? I'm hoping the answer is much easier (than the group cohomology for the general case) in this case where everything is abelian, but I can't find a reference.

Or is it possible to formalize restricting the exact sequences to p-groups for a given prime p?

For example, if the kernel is $C_4^2$, the cokernel is $C_2^2\oplus C_6$, and the group in the middle has exactly 3 minimal generators, I can't think of examples where the group in the middle can be anything but $C_2\oplus C_8\oplus C_{24}$.

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The Ext functor classifies the abelian extensions: $\mathrm{Ext}(A,B)$, in a sense, tells you the equivalence classes of abelian extensions of $A$ by $B$. – Arturo Magidin Jul 2 '12 at 21:09
I have tried reading about this in Dummit and Foote (Section 17.4, where it shows how the extensions are parameterized by H^2(G,A), where A is the kernel and G is the cokernel), but I wondered if somebody might have figured out how the theory of group extensions specializes when everything involved is abelian. – Dtseng Jul 2 '12 at 21:16
Yes; that's what the Ext functor is; $H^2(G,A)$ includes the nonabelian extensions when $G$ and $A$ are both abelian; Ext only deals with the abelian extensions. – Arturo Magidin Jul 2 '12 at 21:18
If the abelian groups are also finite, then yes, there is a relatively clean answer for both Ext() [which repeats many middle groups] and the isomorphism classes themselves. If the abelian groups are general (infinitely generated), then no, there is no nice answer. – Jack Schmidt Jul 2 '12 at 21:23
Yes, the groups are finite (I edited the question). Do you know where I can find this classification? – Dtseng Jul 2 '12 at 21:55
up vote 2 down vote accepted

First of all, we can reduce to $p$-groups. For any finite abelian group $G$, and prime $p$, let $G_p$ be the set of elements of $p$-power of order. Then $G = \prod_p G_p$. We have $\mathrm{Hom}(G, H) \cong \prod_p \mathrm{Hom}(G_p, H_p)$ and one can show (exercise!) that there is a short exact sequence $0 \to A \to B \to C \to 0$ if and only if there is a short exact sequence $$0 \to A_p \to B_p \to C_p \to 0$$ for every $p$.

Finite abelian $p$ groups are classified by partitions: For the partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_r)$ we write $G(p,\lambda) := \prod_{i} \mathbb{Z}/p^{\lambda_i}$.

Let $\lambda$, $\mu$, $\nu$ be three partitions. The following are equivalent:

  1. There is a short exact sequence $$0 \to G(p, \lambda) \to G(p, \nu) \to G(p, \mu) \to 0$$

  2. The Hall polynomial $g_{\lambda \mu}^{\nu}(p)$ is nonzero.

  3. The Littlewood-Richarsdon coefficient $c_{\lambda \mu}^{\nu}$ is nonzero.

  4. Anyone of the zillion equivalent conditions that you can find in, for example, Knutson and Tao or Zelevinsky

For further discussion of this, see Fulton, especially Section 2.

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