Abelian quotient of subgroups of solvable linear groups

Abstract

In this paper, we study the relations of the sizes of abelian quotient of a subgroup of finite linear groups and the largest orbit size of the linear group actions. We study two conjectures raised by Keller and the author. We show that these two conjectures are not true in general, but also prove some special cases under the condition that the group is solvable.