ORBITS OF THE ACTIONS OF ODD-ORDER SOLVABLE GROUPS

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v is an element of V such that C-G(v) subset of F-2(G) (where F-2(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall pi-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v is an element of V such that C-H(v) subset of O-pi(G).