Some families of locally graded groups with finitely many orbits under automorphisms

TL;DR

This paper investigates specific classes of locally graded groups with finitely many automorphism orbits, establishing conditions under which these groups are finite, locally finite, or have finitely many automorphism orbits.

Contribution

It characterizes when certain locally graded groups have finitely many automorphism orbits, including residually finite, finitely generated, and free nilpotent groups.

Findings

Residually finite groups with finitely many automorphism orbits are locally finite and have finite exponent.

Finitely generated locally graded groups with finitely many automorphism orbits are finite.

The Mal'cev ${Q}$-completion of an $r$-generated free nilpotent group has finitely many automorphism orbits only in specific cases.

Abstract

In this work, we study three families of locally graded groups with finitely many orbits under automorphisms. We prove that: (i) a residually finite group with finitely many orbits under automorphisms is locally finite and has finite exponent; (ii) a finitely generated locally graded group with finitely many orbits under automorphisms is finite; and (iii) the Mal'cev $\mathbb{Q}$-completion of an $r$-generated free nilpotent group of class $c$ has finitely many orbits under automorphisms if and only if either $r = 2$ and $c = 3$, or $c \leq 2$