Bounding the composition length of primitive permutation groups and completely reducible linear groups

TL;DR

This paper establishes upper bounds on the composition length of finite permutation and linear groups based on their parameters, providing sharp bounds and extremal examples for various group classes.

Contribution

It introduces new bounds on the composition length of permutation and linear groups, extending understanding of their structural complexity.

Findings

Bounds are sharp in most cases

Extremal examples are characterized

Results apply to primitive, quasiprimitive, and semiprimitive groups

Abstract

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.