Orbits of finite solvable groups on characters

TL;DR

This paper proves that a solvable group acting coprimely on another solvable group has a large orbit on the set of irreducible characters, extending previous results for p-groups with a weaker bound.

Contribution

It generalizes a 2005 result by showing that any solvable group action has a large orbit on characters, not just p-groups, with a weaker bound.

Findings

A solvable group A has a large orbit on G's irreducible characters under coprime action.

Extension of previous p-group results to all solvable groups.

Provides bounds on the size of the orbit in the character set.

Abstract

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a "large" orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This extends (at the cost of a weaker bound) a 2005 result of A. Moreto who obtained such a bound in case that A is a p-group.