# radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or resp. about p-groups?

• Your question is not very specific. I there something you want to know, in particular? Otherwise, refering you to the standard references on the subject is a quite reasonable answer which you will probably not like. – Mariano Suárez-Álvarez Nov 30 '11 at 4:10
• I want to understand the representation theory of p-groups, i.e. kP-modules. There is only one simple kP-module (the trivial module k). But because we are in the modular setting not every kP-module is semisimple, so we also have to study indecomposable modules that are not simple. The projective indecomposable modules are in 1:1 with the simple modules (via the socle) so here we also have only one projective indecomposable module. What i want to understand is how do me the radical layers resp the socle layers help me to understand how kP-modules glue together. – user7475 Nov 30 '11 at 16:49
• «To understand how kP-modules glue together» is 87% of what representation theory is! That is not a specific question, really :) – Mariano Suárez-Álvarez Nov 30 '11 at 17:27