Why is $Rad(P_S)/Soc(P_S)$ a direct sum of two uniserial modules? Where $P$ is the projective cover of some simple $kG$-module $S$. (We may assume char$(k)$|$|G|$.)
This is taken from Page 119 of Alperin's Local Rep Theory.
2026-03-25 07:45:30.1774424730
Why is $Rad(P_S)/Soc(P_S)$ a direct sum of two uniserial modules?
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This isn’t true for general $G$ and $S$, and neither does Alperin claim it is. In the reference you give, this is part of the definition of a Brauer tree algebra. The main result of the rest of the chapter is that block algebras with cyclic defect group are Brauer tree algebras, but block algebras with non-cyclic defect group are not.