Bounding the orders of nilpotent subgroups of solvable linear groups

Abstract

We prove that a nilpotent subgroup $H$ of a finite solvable group $G$ has order at most $|V{|}^{\beta }∕2$ if $V$ is a faithful and completely reducible $G$-module, where $\beta =\text{ln}(32)∕\text{\hspace{0.17em}ln}(9)$. We also find related bounds for nilpotent subgroups of odd order in a solvable linear group. We then further generalize these results to certain chief factors of an arbitrary linear group.