Orbits of $p$-Nilpotent Group Actions for Small Primes

Abstract

Let $G$ be a finite solvable group and $V$ be a faithful completely reducible $G$-module. We improve previous bounds for the sizes of defect groups of blocks of solvable groups through an improved bound on vector stabilizers of $p$-nilpotent groups. Specifically, if $G$ is 2-nilpotent and $2\nmid |V|$, then there exists $v\in V$ such that ${\mathbf{C}}_G(v)\le {\mathbf{F}}_4(G)$. Similarly, if $G$ is 3-nilpotent and $3\nmid |V|$, then there exists $v\in V$ such that ${\mathbf{C}}_G(v)\le {\mathbf{F}}_5(G)$. Using these results, we conclude that $d(B)\le \lfloor \frac{9n}{10}\rfloor$ when $O_p(G)=1$ for $p=2$ and $d(B)\le \lfloor \frac{19n}{21}\rfloor$ when $O_p(G)=1$ for $p=3$.