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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.02090 |
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Table of Contents:
- 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 $c \leq 2$