Linear $2$-Group Actions With Exactly One Regular Orbit
Abstract
Let G be a finite $2$-group and V a finite faithful irreducible G-module. We show that if G has exactly one regular orbit on V, then the characteristic q of V is a Fermat prime, V is irreducible over Bbb Fq, dimBbb FqV is a power of $2$, and the centraliser of G in GL(V) embeds into G. For every Fermat prime q and every m≥ 1 we determine the largest orbit size 2xm of a Sylow $2$-subgroup Pm of GL2m−1(q) on its natural module Vm, and the number of orbits of this size, which is $1$, $2$, or $3$. Our main result is that for every Fermat prime q and every m≥ 1 there is an irreducible subgroup of Pm of order 2xm that has exactly one regular orbit on Vm. In this respect p=2 behaves like p=3, and unlike the Mersenne primes p≥ 7 studied in an earlier paper. The proof places a group related to a Singer cycle into every block of q−1 coordinates. This reduces the problem, for all Fermat primes at once, to a question about subgroups of a Sylow $2$-subgroup of a symmetric group acting on subsets, which we answer by means of fibre products. We also determine which largest orbits occur in two small levels, and we classify the examples in some small cases with the help of a computer.
Related
- Linear Group Actions with Exactly One Regular Orbit
- Finite group actions on the non-minimal nilpotent orbit closures in type A
- Group Actions, Fixed Points and Orbit Quotients in Lipschitz Spaces
- Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space
- The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups
BibTeX
@misc{keller2026linear,
title = {{Linear $2$-Group Actions With Exactly One Regular Orbit}},
author = {Thomas Michael Keller and Jeffrey M. Riedl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27536},
eprint = {2610.27536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27536},
}
RIS
TY - UNPB
TI - Linear $2$-Group Actions With Exactly One Regular Orbit
AU - Thomas Michael Keller
AU - Jeffrey M. Riedl
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27536
UR - https://arxiv.org/abs/2610.27536
AB - Let G be a finite $2$-group and V a finite faithful irreducible G-module. We show that if G has exactly one regular orbit on V, then the characteristic q of V is a Fermat prime, V is irreducible over Bbb F_q, dim_Bbb F_qV is a power of $2$, and the centraliser of G in GL(V) embeds into G. For every Fermat prime q and every m≥ 1 we determine the largest orbit size 2^x_m of a Sylow $2$-subgroup P_m of GL_2^m−1(q) on its natural module V_m, and the number of orbits of this size, which is $1$, $2$, or $3$. Our main result is that for every Fermat prime q and every m≥ 1 there is an irreducible subgroup of P_m of order 2^x_m that has exactly one regular orbit on V_m. In this respect p=2 behaves like p=3, and unlike the Mersenne primes p≥ 7 studied in an earlier paper. The proof places a group related to a Singer cycle into every block of q−1 coordinates. This reduces the problem, for all Fermat primes at once, to a question about subgroups of a Sylow $2$-subgroup of a symmetric group acting on subsets, which we answer by means of fibre products. We also determine which largest orbits occur in two small levels, and we classify the examples in some small cases with the help of a computer.
ER -