Automorphism Orbits of the Group of Unitriangular Matrices
Emerson de Melo, Júlia Kato
TL;DR
This work studies automorphism orbits of unitriangular groups $UT_n(\mathbb{F})$ over an infinite field $\mathbb{F}$ by examining the action of $\mathrm{Aut}(UT_n(\mathbb{F}))$ on the group itself. It leverages the known generating set and semi-direct product structure of $\mathrm{Aut}(UT_n(\mathbb{F}))$, together with a partition-and-representative strategy, to count orbits for small $n$ and to prove infinitude for larger $n$. The main results are sharp: $\omega(UT_n(\mathbb{F}))$ is finite for $n\le 5$ with explicit counts $\omega(UT_3(\mathbb{F}))=3$, $|\omega|=16$ for $n=4$, and $|\omega|=61$ for $n=5$ (the latter obtained with SageMath). For $n\ge 6$, the paper proves $\omega(UT_n(\mathbb{F}))$ is infinite, using a construction modulo $\gamma_4$ and a reduction to field automorphisms, then extending by embedding to larger $n$. These results delineate a clear boundary in automorphism-orbit behavior and contribute to the broader understanding of automorphism structures in nilpotent linear groups.
Abstract
Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called the automorphism orbits of $G$, and their number is denoted by $ω(G)$. Let $\mathbb{F}$ be an infinite field, and let $UT_n(\mathbb{F})$ denote the group of unitriangular matrices over $\mathbb{F}$. We show that $ω(UT_n(\mathbb{F}))$ is finite for $n \leq 5$ and infinite for $n \geq 6$.