Linear Group Actions with Exactly One Regular Orbit
Abstract
Let p be an odd prime, G a finite p-group, and V a finite faithful irreducible G-module. We show that if G has exactly one regular orbit on V, then p=2n−1 is a Mersenne prime and (G,V) embeds into (Pm,Vm) for some m≥ 1, where Pm is a Sylow p-subgroup of the monomial group GL1(2n)wr Spm−1 and Vm is its natural module, regarded over Bbb F2. We determine the largest orbit size pxm of Pm on Vm and the number of orbits of this size, and we construct subgroups of Pm of order pxm whose unique regular orbit is a largest orbit of Pm; for p=3 such subgroups exist for every m, whereas for Mersenne primes p≥ 7 we prove that they exist only for m≥ 3. Finally, we classify the subgroups of Pm of order pxm with exactly one regular orbit for m=2 and, with the help of a computer, for p=3 and m=3, where there are $13$ such pairs up to isomorphism; for p=3 and m=4 we show that there are exactly $4914$ conjugacy classes of such subgroups in P4. In particular, such a subgroup is in general not determined by its regular orbit.
Related
- Linear $2$-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 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.2610.25737},
eprint = {2610.25737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25737},
}
RIS
TY - UNPB
TI - Linear Group Actions with Exactly One Regular Orbit
AU - Thomas Michael Keller
AU - Jeffrey M. Riedl
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25737
UR - https://arxiv.org/abs/2610.25737
AB - Let p be an odd prime, G a finite p-group, and V a finite faithful irreducible G-module. We show that if G has exactly one regular orbit on V, then p=2^n−1 is a Mersenne prime and (G,V) embeds into (P_m,V_m) for some m≥ 1, where P_m is a Sylow p-subgroup of the monomial group GL_1(2^n)wr S_p^m−1 and V_m is its natural module, regarded over Bbb F_2. We determine the largest orbit size p^x_m of P_m on V_m and the number of orbits of this size, and we construct subgroups of P_m of order p^x_m whose unique regular orbit is a largest orbit of P_m; for p=3 such subgroups exist for every m, whereas for Mersenne primes p≥ 7 we prove that they exist only for m≥ 3. Finally, we classify the subgroups of P_m of order p^x_m with exactly one regular orbit for m=2 and, with the help of a computer, for p=3 and m=3, where there are $13$ such pairs up to isomorphism; for p=3 and m=4 we show that there are exactly $4914$ conjugacy classes of such subgroups in P_4. In particular, such a subgroup is in general not determined by its regular orbit.
ER -