Character induction in p-groups (Q1084172)

Let G be a finite p-group, \(\chi\in Irr(G)\), \(\Phi(G)\) the Frattini subgroup of G, \(V(\chi)=<g\in G: \chi (g)\neq 0>\), and if \(N\triangleleft G\), \(\psi\in Irr(N)\), \(I_ G(\Psi)=\{g\in G:\psi^ g=\psi \}\). The main result of this paper is: Let G be a finite p-group, \(\chi\in Irr(G)\), \(\chi\) non-linear, \(N\triangleleft G\) with \(V(\chi) \leq N \leq V(\chi)\Phi(G)\), and \(\psi\) an irreducible constituent of \(\chi_ N\). If \(\psi\) is non-linear then \(I_ G(\psi)<G\). Using this result, the author forms chains of subgroups with associated characters each of which induces \(\chi\).