Inductive sources and subnormal subgroups.
By a character pair in a finite group \(G\) is meant a pair \((H,\theta)\), where \(H\leq G\) and \(\theta\in\text{Irr}(H)\). The group \(G\) acts on the set of character pairs by \((H,\theta)^g=(H^g,\theta^g)\), where \(g\in G\). The character \(\theta^g\) of \(H^g\) is defined by the formula \(\theta^g(h^g)=\theta(h)\) for \(h\in H\). The stabilizer of \((H,\theta)\) in this action coincides with the inertia group of \(\theta\) in \(N_G(H)\). The character pair \((H,\theta)\) is an `inductive source' for \(G\), if induction defines an injection from \(\text{Irr}(T|\theta)\) into \(\text{Irr}(G)\), where \(T\) is the stabilizer of \((H,\theta)\) in \(G\). If, for example, \(H\) is normal in \(G\), then \((H,\theta)\) is an inductive source for \(G\). On the other hand, if \(H\) is not normal in \(G\), then the pair \((H,1_H)\) is never an inductive source. In this note a necessary condition for a character pair \((H,\theta)\) to be an inductive source is given. It appears that this condition is also sufficient in the case where \(H\) is subnormal in \(G\). We write \((H,\theta)\leq(K,\psi)\) if \(H\leq K\) and \(\langle\theta,\psi_H\rangle>0\). We say that a character pair \((H,\theta)\) is `conjugate stable' in \(G\), if whenever there is a pair \((K,\psi)\) in \(G\) and \(g\in G\) such that both \((H,\theta)\leq(K,\psi)\) and \((H,\theta)^g\leq(K,\psi)\), then there is \(k\in K\) such that \((H,\theta)^g=(H,\theta)^k\). Theorem A. Let \((H,\theta)\) be an inductive pair in a group \(G\). Then \((H,\theta)\) is conjugate stable in \(G\).
scientific article; zbMATH DE number 2113839