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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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    @mt_: Thanks. I found the answer in Benson chapter 1, p. 12, in the discussion under Corollary 1.7.4.2013-05-11

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Following the comments, I left an answer as part of the following question: https://math.stackexchange.com/questions/388033/understanding-a-paper-concerning-the-a-group-ring-an-augmentation-ideal-and-the