Normal subgroups and multiplicities of indecomposable modules
Let \(R\) be an algebraically closed field of characteristic \(p\neq 0\) or a suitable \(p\)-adic valuation ring. Moreover, let \(N\) be a normal subgroup of a finite group \(G\), and let \(M\) be an indecomposable \(RG\)-module with vertex \(Q\). The paper under review is concerned with the multiplicity \(n\) of an indecomposable direct summand \(V\) of \(M_N\) and with the connection between the heights of \(M\) and \(V\). One of the main results shows that, if \(V\) is \(G\)-stable, then \(n\) is divisible by the \(p\)-part of \(|G:QN|\). This generalizes the well known fact that \(\text{rk}_R(M)\) is divisible by the \(p\)-part of \(|G:Q|\). The author also shows that the intersection of the kernels of all irreducible characters of height 0 in a \(p\)-block \(B\) of \(G\) with defect group \(D\) is the largest normal subgroup of \(G\) contained in \(\text{Ker}(B)D'\); in particular, it coincides with \(\text{Ker}(B)\) in case \(D\) is abelian.
scientific article; zbMATH DE number 1018455