Finite power image families and automorphism actions in linear groups
Abstract
Let k/ℚ be a finitely generated field extension. We characterize the subgroups G≤ GLn(k) for which the power images Pm(G)={gm:g∈ G} form a finite family: this holds precisely when G=Nrtimes F, where F is finite and N is a uniquely divisible subgroup consisting of unipotent matrices. The subgroup N is canonical and fully invariant.For e=exp(G/N), we obtain Pm(G)=Pgcd(m,e)(G), exactly τ(e) distinct power images, and least positive period e, where τ(e) is the number of positive divisors of e.Over ℚ, the group coincides with the full group of rational points of its Zariski closure. Since finite automorphism-orbit count implies finiteness of the power-image family, an affirmative answer to Problem~21.40 of the Kourovka Notebook\cite{Kourovka} follows as a corollary. We further prove that a rational linear group G admits a decomposition G≅(ℚd,+)rtimes F, with d≥0 finite and F finite, whenever a virtually nilpotent subgroup of Aut(G) has finitely many orbits on G.The latter argument uses nonconstant invariant rational functions arising from nonzero Lie brackets to obstruct dense orbits. A metabelian action with five orbits on the rational Heisenberg group shows that solubility of the acting group is insufficient.
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BibTeX
@misc{jiang2026finite,
title = {{Finite power image families and automorphism actions in linear groups}},
author = {Zhexin Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23373},
eprint = {2610.23373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite power image families and automorphism actions in linear groups
AU - Zhexin Jiang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23373
UR - https://arxiv.org/abs/2610.23373
AB - Let k/ℚ be a finitely generated field extension. We characterize the subgroups G≤ GL_n(k) for which the power images P_m(G)={g^m:g∈ G} form a finite family: this holds precisely when G=Nrtimes F, where F is finite and N is a uniquely divisible subgroup consisting of unipotent matrices. The subgroup N is canonical and fully invariant.For e=exp(G/N), we obtain P_m(G)=P_gcd(m,e)(G), exactly τ(e) distinct power images, and least positive period e, where τ(e) is the number of positive divisors of e.Over ℚ, the group coincides with the full group of rational points of its Zariski closure. Since finite automorphism-orbit count implies finiteness of the power-image family, an affirmative answer to Problem~21.40 of the Kourovka Notebook\cite{Kourovka} follows as a corollary. We further prove that a rational linear group G admits a decomposition G≅(ℚ^d,+)rtimes F, with d≥0 finite and F finite, whenever a virtually nilpotent subgroup of Aut(G) has finitely many orbits on G.The latter argument uses nonconstant invariant rational functions arising from nonzero Lie brackets to obstruct dense orbits. A metabelian action with five orbits on the rational Heisenberg group shows that solubility of the acting group is insufficient.
ER -