Regular orbits of finite primitive solvable groups, III

TL;DR

This paper investigates the structure of finite solvable groups acting on vector spaces and establishes conditions under which these groups have regular orbits, focusing on specific parameters related to extraspecial p-groups.

Contribution

It characterizes the normal subgroup structure of such groups and proves the existence of regular orbits for particular values of a key parameter.

Findings

Regular orbits exist for e=2,3,4,8,9,16 under certain conditions.

The normal subgroup E is uniquely determined and is a product of extraspecial p-groups.

The size of the vector space influences the existence of regular orbits.

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 $e=\sqrt{|E/\bZ(E)|}$. We prove that when $e=2,3,4,8,9,16$, $G$ will have regular orbits on $V$ when the corresponding vector space is not too small.