**Notation:** For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.


**Problem set up:**


Consider, for a finite group $G$, and $n > 1$, the set $Q(G)\_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H\_{n-1}}$, for any normal subgroup $H\_{n-1}$ of $G^n$ isomorphic to $G^{n-1}$.


Note that $Q(G)\_1 \subset Q(G)\_2 \subset \cdots$, and so for finite groups $G$, the sequence stabilises at some $N$ - i.e. we have $Q(G)\_n =Q(G)\_{n+1}$ for all $n \geq N$. We call the sequence $Q(G)\_1, \cdots, Q(G)\_N$ the *quotient series* of $G$.


We recall that by the classification theorem for finite abelian groups, we can write any finite abelian group as $\bigoplus\_{i = 0}^M \mathbb Z\_{p\_{i}^{k\_i}}$ for primes $p\_i$ and positive integers $k\_i$.



> 
> **Question:** Let $G = \bigoplus\_{i = 0}^M \mathbb Z\_{p\_{i}^{k\_i}}$ be a finite abelian group. What is the quotient series of $G$?
> 
> 
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Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$
I'll abbreviate $\mathbb{Z}/p^k \mathbb{Z}$ to $L\_k$. Let $G = \bigoplus\_{i=1}^M L\_i^{a\_i}$. I claim that $\bigoplus\_{i=1}^N L\_i^{b\_i}$ is in the quotient series of $G$ if and only if the Littlewood-Richardson coefficient
$$c\_{(1^a\_1),\ \ (2^{a\_2}),\ \ \dotsc,\ \ (M^{a\_M})}^{(1^{b\_1} 2^{b\_2} \cdots N^{b\_N})}$$
is nonzero. Note the parentheses and commas: We have $M$ distinct partitions on the bottom and only one partition on the top.


You can decide whether or not this is an explicit enough answer to be useful to you.




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**Proof**. In general, let $\alpha$, $\beta$ and $\gamma$ be three partitions. Then there is a short exact sequence
$$0 \to \bigoplus L\_{\alpha\_i} \to \bigoplus L\_{\beta\_i} \to \bigoplus L\_{\gamma\_i} \to 0$$
if and only if $c\_{\alpha \gamma}^{\beta}$ is nonzero. This is the same as asking that there be a semistandard Young tableaux of shape $\beta/\alpha$ whose rectification is a given tableaux of shape $\gamma$. See, for example, Fulton's survey [Eigenvalues, invariant factors, highest weights, and Schubert calculus](https://arxiv.org/abs/math/9908012). The short exact sequences of abelian groups are in Section 2; the tableaux are briefly mentioned in Section 9 and can be found in more detail in many places, such as Stanley's *[Enumerative Combinatorics II](https://doi.org/10.1017/CBO9780511609589)* or Fulton's *[Young Tableaux](https://doi.org/10.1017/CBO9780511626241)*.


In your situation, you want to know when we have $c\_{T \alpha,\ \gamma}^{(T+1) \alpha}$ nonzero for $T$ large. So we are looking for tableaux of shape $(T+1)\alpha/T \alpha$ whose rectification is a given tableaux of shape $\gamma$. But, once $T$ is large enough, $(T+1)\alpha/T \alpha$ is just a union of disconnected rectangles. Specifically, if $\alpha = 1^{a\_1} 2^{a\_2} \cdots M^{a\_M}$, then the connected components of $(T+1)\alpha/T \alpha$ are rectangles of shape $k \times a\_k$, for $1 \leq k \leq M$. Standard properties of Young tableaux then give the answer above.


