Let:
- $G$ be a finite group;
- $p$ be prime;
- $J$ 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 $\mathbb{F}_pG$.
- Why does this object exist?
- Why is it unique?
- 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.