Normal subgroups and heights of characters
This paper deals with relations between the blocks of a finite group \(G\) and those of a normal subgroup \(N\) resp. of the factor group \(\overline G=G/N\). The first part of the paper is devoted to the notation ``domination'', which means: Given an indecomposable \(OG\)-module \(V\) (\(O\) a large field of characteristic \(p>0\) or the ring of integers in a \(p\)-modular system for \(G\)) which is indecomposable as an \(ON\)-module, then a (\(p\)-)block \(B\) of \(G\) dominates a block \(\overline{B}\) of \(\overline G\) through \(V\), if there exists an \(O\overline G\)-module \(X\) in \(\overline B\) such that \(V\otimes\text{infl}(X)\) belongs to \(B\). (\(\text{infl}(X)\) means the inflation of \(X\) to \(G\).) This is equivalent, as the author shows, to the fact that \(B\) covers the block \(b\) of \(ON\) to which \(V_N\) belongs. Furthermore each block \(\overline B\) of \(\overline G\) is dominated through \(V\) by a unique block \(B\) of \(G\) and has a defect group contained in \(DN/N\) where \(D\) is a defect group of \(B\). In the latter statement equality holds for some block \(\overline B\) which is dominated by \(B\) through \(V\). Applications are given when \(V\) affords an irreducible character (ordinary or Brauer) which remains irreducible as a character of \(N\). In the second part of the paper which relies on some of the investigations in the first part, the author investigates heights of characters. He mainly proves: Let \(\xi=\text{Irr}(N)\) belong to a block \(b\) of \(N\) and let \(B\) be a block of \(G\) with defect group \(D\) covering \(b\). Furthermore, let \(\widehat{b}\) be the unique block of \(DN\) covering \(b\) and let \(T_G(b)\) be the inertial group of \(b\) in \(G\). Then the following three numbers are equal: (i) \(\min\{\text{ht}(\chi)-\text{ht}(\xi)\mid\chi\in\text{Irr}(B)\) lying over \(\xi\}\), (ii) \(\min\{\text{ht}(\zeta)-\text{ht}(\xi)\mid\zeta\in\text{Irr}(\widehat b)\) lying over a \(T_G(b)\)-conjugate of \(\xi\}\), (iii) \(\min\{d(B)-\nu(|Q|)\mid Q\leq D\) such that \(\xi^t\) extends to \(QN\) for some \(t\in T_G(b)\}\), where \(d(B)\) denotes the defect of \(B\) and \(\nu(|Q|)\) the \(p\)-part of \(|Q|\). A similar result holds for Brauer characters. As applications the author gives a slight improvement of a result of Gluck-Wolf and proves one of the Dade conjectures in a very special situation.
scientific article; zbMATH DE number 950577