Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on vector space V, is not metacyclic. Then always has regular orbit V except for few “small” cases.

Let $G$ be a finite primitive permutation group on set $\Omega$ with point stabiliser $H$. Recall that subset of is base for if its pointwise trivial. We define the size $G$, denoted $b(G,H)$, to minimal $G$. Determining fundamental problem in theory, long history stretching back 19th century. Here one our main motivations theorem Seress from 1996, which states $b(G,H) \leqslant 4$ soluble. In ...