Classifying groups with three automorphism orbits

Abstract: We call a group $G$ a $k$-orbit group if its automorphism group $Aut(G)$ acting naturally on $G$ has precisely $k$ orbits. I will describe the classification of finite 3-orbit groups after surveying work to classify $k$-orbit groups for small $k$ when $G$ is finite or infinite. The finite 3-orbit groups that are not $p$-groups are easy to classify. Apart from $Q_8$, the finite non-abelian 3-orbit 2-groups are a subset of the Suzuki 2-groups which Graham Higman [2] classified in 1963. Determining which subset turns out to be far from easy as the automorphism groups of Suzuki 2-groups are mysterious. Alex Bors and I classified the finite 3-orbit 2-groups in [1]. In 2024 Li and Zhu [3], unaware of our work and using different methods, classified the finite groups $G$ where $Aut(G)$ is transitive on elements of order $p$. Their groups include the 3-orbit Suzuki 2-groups, the homocyclic groups $C_{p^n}^m$ of exponent $p^2$ and the generalised quaternion group $Q_{2^{n+1}}$.

I was able to classify all finite 3-orbit groups (including $p>2$) using Hering's Theorem and some representation theory. However, to my surprise Li and Zhu [4] in March 2024 did the same.

[1] Alexander Bors and S.P. Glasby, Finite 2-groups with exactly three automorphism orbits, arxiv.org/abs/2011.13016v1 (2020).

[2] G. Higman, Suzuki 2-groups, Illinois J.~Math. 7 (1963), 79--96.

[3] Cai Heng Li and Yan Zhou Zhu, A Proof of Gross' Conjecture on 2-Automorphic 2-Groups, arxiv.org/abs/2312.16416 (2024).

[4] Cai Heng Li and Yan Zhou Zhu, The finite groups with three automorphism orbits, arxiv.org/abs/2403.01725 (2024).

group theory

Audience: researchers in the discipline

Symmetry in Newcastle