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PreprintCC BY-NC-ND 4.0arXiv

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 ∈ V such that CG(v) ≤ F4(G). Similarly, if G is 3-nilpotent and 3 nmid |V|, then there exists v ∈ V such that CG(v) ≤ F5(G). Using these results, we conclude that d(B) ≤ ⌊ 9n/10 ⌋ when Op(G) = 1 for p = 2 and d(B) ≤ ⌊ 19n/21 ⌋ when Op(G) = 1 for p = 3.

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Cite

BibTeXRIS
BibTeX
@misc{so2026orbits,
  title = {{Orbits of $p$-Nilpotent Group Actions for Small Primes}},
  author = {Chloe So and Yong Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30952},
  eprint = {2610.30952},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30952},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Orbits of p-Nilpotent Group Actions for Small Primes
AU  - Chloe So
AU  - Yong Yang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30952
UR  - https://arxiv.org/abs/2610.30952
AB  - 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 ∈ V such that C_G(v) ≤ F_4(G). Similarly, if G is 3-nilpotent and 3 nmid |V|, then there exists v ∈ V such that C_G(v) ≤ F_5(G). Using these results, we conclude that d(B) ≤ ⌊ 9n/10 ⌋ when O_p(G) = 1 for p = 2 and d(B) ≤ ⌊ 19n/21 ⌋ when O_p(G) = 1 for p = 3.
ER  -