Regular orbits of finite primitive solvable groups

Abstract

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V . Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p -groups for various p and we denote e = | E / Z ( E ) | . We prove that when e ⩾ 10 and e ≠ 16 , G will have at least 5 regular orbits on V . We also construct groups with no regular orbits on V when e = 8 , 9 and 16.