Abelian quotients and the largest orbit sizes of linear groups of odd order

Abstract

Let G be a finite nonabelian group of odd order, let p be the smallest prime divisor of |G|, and let V be a finite faithful completely reducible G-module, possibly of mixed characteristic. Suppose that M is the largest orbit size in the action of G on V. It is known that |G:G'|≤ M, and that equality holds only for abelian groups if |G| is odd. We prove that |G:G'|≤ M/p, unless p=3 and G is the direct product of an abelian group and a group isomorphic to Γ(2³) or to Γ(2³)× C₃ acting on V in a specific way, in which case |G:G'|=3M/7. In particular, |G:G'|≤ 3M/7 for every nonabelian group of odd order. If |V| is odd, then |G:G'|=M/p holds if and only if |G'|=p; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy |G:G'|≤ 3M/13, and even |G:G'|<M/(2p) if p≥5. The bound 3M/13 is best possible.

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BibTeX RIS

BibTeX

@misc{keller2026abelian,
  title = {{Abelian quotients and the largest orbit sizes of linear groups of odd order}},
  author = {Thomas Michael Keller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08690},
  eprint = {2610.08690},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08690},
}

RIS

TY  - UNPB
TI  - Abelian quotients and the largest orbit sizes of linear groups of odd order
AU  - Thomas Michael Keller
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08690
UR  - https://arxiv.org/abs/2610.08690
AB  - Let G be a finite nonabelian group of odd order, let p be the smallest prime divisor of |G|, and let V be a finite faithful completely reducible G-module, possibly of mixed characteristic. Suppose that M is the largest orbit size in the action of G on V. It is known that |G:G'|≤ M, and that equality holds only for abelian groups if |G| is odd. We prove that |G:G'|≤ M/p, unless p=3 and G is the direct product of an abelian group and a group isomorphic to Γ(2^3) or to Γ(2^3)× C_3 acting on V in a specific way, in which case |G:G'|=3M/7. In particular, |G:G'|≤ 3M/7 for every nonabelian group of odd order. If |V| is odd, then |G:G'|=M/p holds if and only if |G'|=p; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy |G:G'|≤ 3M/13, and even |G:G'|<M/(2p) if p≥5. The bound 3M/13 is best possible.
ER  -

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