On extensions of projective indecomposable modules. (Q818020)

Let \(p\) be a prime, let \(G\) be a finite group and let \((K,R,k)\) be a \(p\)-modular system that is large enough for~\(G\). That is, \(R\) is a complete discrete valuation ring with \(k=R/J(R)\) an algebraically closed field of characteristic \(p\) and with~\(K\), the field of fractions of \(R\), of characteristic \(0\) such that \(K\) contains a primitive \(|G|\)-th root of unity. Assume that \(N\) is a normal subgroup of \(G\) with a \(G\)-stable block \(b\) of \(RN\) having a defect group \(Q\) such that \(N=QC_N(Q)\). Let \(V\) be a unique-up-to-isomorphism projective indecomposable \(b\)-module. Assume also that \(G/N\) is a \(p\)-group and let \(B\) be the unique block of \(G\) that covers~\(b\). The main results of this paper are: Proposition. Suppose that \(U\) is an extension of \(V\) to \(RG\) and let \(P\) be a vertex of~\(V\) with \(P\)-source \(W\). Then (a)~\(PQ\) is a defect group of~\(B\); and (b)~If \(P\cap N=1\) also, then \(W\) is an endo-permutation module which is identified with a lift of a source of a unique-up-to-isomorphism simple \(kG\)-module in~\(B\). Also if \(G/N\) is cyclic, then any indecomposable \(RG\)-module in~\(B\) with vertex \(P\) and \(P\)-source \(W\) is an extension of~\(V\).