Cartan invariants of group algebras of finite groups 

Shigeo Koshitani

Proc. Amer. Math. Soc. 124 (1996), pp. 2319-2323 

Abstract. We give a result on Cartan invariants of the group algebra kG of a 
finite group G over an algebraically closed field k, which implies that if the 
Loewy length (socle length) of the projective indecomposable kG-module 
corresponding to the trivial kG-module is four, then k has characteristic 2. 
The proof is independent of the classification of finite simple groups.