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Let:

  1. $G$ be a finite group;
  2. $p$ be prime;
  3. $J$ be the Jacobson radical of $\mathbb{F}_pG$.

A paper I'm trying to read mentions the following object:

The indecomposable projective $\mathbb{F}_pG$-module $U$ with $U/UJ\cong\mathbb{F}_p$

It is then also claimed that $U$ is a direct summand of $\mathbb{F}_pG$.

  1. Why does this object exist?
  2. Why is it unique?
  3. Why is it a direct summand of $\mathbb{F}_pG$.

I know all the definitions of the terms mentioned, but not experienced with some of them.

The paper is "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)).

EDIT: If relevant, it might be understood from context that $p$ divides $|G|$, but I'm nore sure.

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Since $\mathbb{F}_p[G]$ is Artinian, $\mathbb{F}_p[G]/J$ is semisimple Artinian. Hence every submodule of $\mathbb{F}_p[G]/J$ is a direct summand. In particular $\mathbb{F}_p\subseteq \mathbb{F}_p[G]/J$ is the image of an idempotent $e\in \mathbb{F}_p[G]/J$. By the descending chain condition and Nakayama's lemma, $J$ is nilpotent. Hence we can lift $e$ to an idempotent $\hat{e}\in \mathbb{F}_p[G]$. The image of $\hat{e}$ is probably the module you want. Unfortunately, it isn't clear how this approach gives uniqueness. –  Jeff Tolliver Nov 21 '12 at 1:26
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Alternatively: every simple module $S$ has a projective cover $P$ which is a direct summand of $\mathbb{F}_pG$, and is such that $P/PJ \cong S$. Furthermore the multiplicity of $P$ as a direct summand of $\mathbb{F}_pG$ is $\dim S$. I'd recommend you read something like Benson - Representations and Cohomology I or Alperin - Local representation theory or one of the Curtis-Reiner books to learn the relevant theory. –  mt_ Nov 21 '12 at 10:29
    
@mt_: Sorry for the long delay in my reply. Can you point me to the place in one of those books which explains the relevant material to understand your answer? –  user3533 May 6 '13 at 8:31
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@user3533 Benson chapters 1 (background on the representation theory of algebras, Wedderburn's theorem, existence and uniqueness of projective covers, projectives are summands of free modules,...) and 3 (group representations). –  mt_ May 6 '13 at 8:47
    
@mt_: Thanks. I found the answer in Benson chapter 1, p. 12, in the discussion under Corollary 1.7.4. –  user3533 May 11 '13 at 23:10
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Following the comments, I left an answer as part of the following question: Understanding a paper concerning the a group-ring, an augmentation ideal and the Jacobson radical

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