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Let $A$ be an abelian group and $G$ be a group. A short exact sequence of groups like $1\longrightarrow A\longrightarrow E\longrightarrow G\longrightarrow 1$ is called an extension. We say that $E$ is an extension of $A$ by $G$. This extension makes $A$ into a $G$-module.

Assume that $A$ is an elementary abelian p-subgroup of rank $m$ and $G$ be an elementary abelian p-subgroup of rank $2$. Then $A$ is a $F_{p}[G]$ -module with $F_{p}$ is a finite field of $p$ elements. Obviously, this structure of $F_{p}[G]$-module is not unique. Thus, one can ask the following question:

Question: What is the number of structure of $F_{p}[G]$-modules defined on $A$?.

Any help would be appreciated so much. Thank you all.

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  • $\begingroup$ I don't see how the extension gives to $A$ a $G$-module structure. How do you define the action of $G$? $\endgroup$
    – Antoine Labelle
    Commented Aug 17, 2020 at 2:16
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    $\begingroup$ @Antoine: the conjugation action of $E$ on $A$ factors through $G$ because $A$ is abelian. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 17, 2020 at 5:23
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    $\begingroup$ Dos this question have anything to do with group extensions? You seem to be just asking for the number of isomorphism classes of ${\mathbb F}_pG$-modules over ${\mathbb F}_p$, where $G$ is elementary abelian of order $p^2$. That is a question in representation theory. $\endgroup$
    – Derek Holt
    Commented Aug 17, 2020 at 7:24
  • $\begingroup$ Yes, exactly. Thank you for your clarification. Can you explain why that is a question in representation theory. Are there similar questions in representation theory. $\endgroup$
    – Nourddine Snanou
    Commented Aug 17, 2020 at 16:49
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    $\begingroup$ @Nourddine Snanou Group representations of group $G$ over a field $K$ are the same as $K[G]$-modules and this is exaftly what representation theory studies. In your case $K$ is $\mathbb{F}_p$ and $G\cong (\mathbb{Z}/p\mathbb{Z})^2$. Since the characteristic of $K$ divides $|G|$, Maschke's theorem does not hold, so this is more precisely a question of modular representation theory. I don't know much about modular representation theory, though, so I can't help you more. $\endgroup$
    – Antoine Labelle
    Commented Aug 18, 2020 at 19:53

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There is a large number of distinct (up to module isomorphism) ${\mathbb F}_pG$-modules, even in small dimensions, and I don't think you can hope for any kind of reasonable classification.

I did some quick computer calculations. For $p=2$, the number of isomorphism classes of dimensions 1,2, …, 7 is 1, 5, 11, 28, 53, 111, 199.

For $p=3$ in dimensions 1,2,3,4,5, it is 1,6, 25, 78, 235.

For $p=5$ in dimensions 1,2,3,4, it is 1,8,47,310.

Of course, it is sufficient to classify the indecomposable modules, but ${\mathbb F}_pG$ is of so-called wild representation type, which means more or less that this cannot easily be done!

For example, with $p=5$ in dimension 4, 242 of the 310 modules are indecomposable.

On the other hand, I am by no means an expert in modular representation theory, and it is possible that more can be said.

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