Automorphism Orbits of the Group of Unitriangular Matrices

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 $\omega(G)$. 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 $\omega(UT_n(\mathbb{ F}))$ is finite for $n \leq 5$ and infinite for $n \geq 6$.

Categories: math.GR