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arXiv · 2610.30952

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

Chloe So·Yong Yang
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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 $\mathbf{C}_G(v) \leq \mathbf{F}_4(G)$. Similarly, if $G$ is 3-nilpotent and $3 \nmid |V|$, then there exists $v \in V$ such that $\mathbf{C}_G(v) \leq \mathbf{F}_5(G)$. Using these results, we conclude that $d(B) \leq \lfloor \frac{9n}{10} \rfloor$ when $O_p(G) = 1$ for $p = 2$ and $d(B) \leq \lfloor \frac{19n}{21} \rfloor$ when $O_p(G) = 1$ for $p = 3$.

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Chloe So, Yong Yang. 2026-10-22. Orbits of $p$-Nilpotent Group Actions for Small Primes. https://arxiv.org/abs/2610.30952

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