Class 2 quotients of solvable linear groups

TL;DR

This paper generalizes a known bound on the order of nilpotent class 2 groups acting on modules, extending it to solvable groups and their maximal class 2 quotients, showing they are strictly bounded by the module size.

Contribution

It proves that for solvable groups, the order of the maximal class 2 quotient is strictly less than the size of a faithful, completely reducible module.

Findings

Maximal class 2 quotient order is strictly less than module size

Generalizes Glauberman's result from nilpotent to solvable groups

Provides bounds on group quotients in relation to module dimensions

Abstract

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. By a result of Glauberman it has been known for a long time that if $G$ is nilpotent of class 2, then $|G| < |V|$. In this paper we generalize this result as follows. Assuming $G$ to be solvable, we show that the order of the maximal class 2 quotient of $G$ is strictly bounded above by $|V|$.