On a short exact sequence and extensions of irreducible \(\mathbb{K} G\)-modules (Q5939615)

Let \(G\) be a finite group and let \(K=\mathbb{F}_p\) be the prime field of characteristic \(p\). For a simple \(KG\)-module \(U\) let \(P_G(U)\) denote the projective cover of \(U\) and \(L_2(U)=P_G(U)J/P_G(U)J^2\) the second Loewy-layer of \(P_G(U)\), where \(J\) is the Jacobson radical of KG. If \(U=K\) is the trivial module and \(G\) is \(p\)-solvable, Gaschütz completely described \(L_2(K)\) in terms of \(G\), i.e., \(L_2(K)\) consists of the complemented \(p\)-chief factors of \(G\) counted with multiplicities. In the paper under review the case of a general simple \(U\) is investigated. Under certain assumptions the author constructs a short exact sequence from the inflation restriction sequence which relates to \(L_2(U)\). She shows that the assumptions are satisfied if \(U\) is projective as a \(G/C_G(U)\)-module. In this case and if in addition \(G\) is \(p\)-solvable she describes \(L_2(U)\) as a direct sum of heads of tensor products \(U\otimes(F_i)_K\) where the \(F_i\) are \(p\)-chief factors of \(G\) which lie below \(C_G(U)\) and are complemented in \(C_G(U)\). An illumining example shows that the assumptions which have to be made to arrive at the short exact sequence can not be dropped.