The existence of regular orbits
Let G be a solvable group and p an odd prime. It is shown that if A is a p-subgroup of G and X is a p-chief factor of G on which A acts faithfully, then X contains a regular A-orbit, i.e., there exists \(x\in X\) such that \(C_ A(x)=1\). The result is false for \(p=2\). As a corollary, the following character-theoretic result is obtained: Let G be a solvable primitive group. Assume that F(G) is a p-group, p odd. Take \(P\in S_ p(G).Then\) \(b(P)=\max \{\psi (1)|\) \(\psi \in Irr(P)\}=| G:F(G)|_ p\).
scientific article; zbMATH DE number 4067171