BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Stephen Glasby (University of Western Australia) DTSTART:20240320T010000Z DTEND:20240320T020000Z DTSTAMP:20260423T053047Z UID:SiN/58 DESCRIPTION:Title: Cla ssifying groups with three automorphism orbits\nby Stephen Glasby (Uni versity of Western Australia) as part of Symmetry in Newcastle\n\nLecture held in VG10.\n\nAbstract\nWe call a group $G$ a $k$-orbit group if its a utomorphism group $Aut(G)$ acting naturally on $G$ has \nprecisely $k$ orb its. I will describe the classification of finite 3-orbit groups after sur veying\nwork 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 t o 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 196 3. Determining which subset turns out to be far from easy as the automorph ism groups of Suzuki 2-groups are mysterious.\nAlex Bors and I classified the finite 3-orbit 2-groups in [1]. In 2024 Li and Zhu [3]\, unaware of ou r work\nand using different methods\, classified the finite groups $G$ whe re $Aut(G)$ is transitive on elements of order $p$. Their groups include t he 3-orbit Suzuki 2-groups\, the homocyclic groups $C_{p^n}^m$ of exponent $p^2$ and the generalised quaternion group $Q_{2^{n+1}}$.\n\nI was able t o classify all finite 3-orbit groups (including $p>2$) using Hering's Theo rem and some representation theory. However\, to my surprise Li and Zhu [4 ] in March 2024 did the same.\n\n[1] Alexander Bors and S.P. Glasby\,\nFin ite 2-groups with exactly three automorphism orbits\, https://arxiv.org/ab s/2011.13016v1 (2020).\n\n[2] G. Higman\, Suzuki 2-groups\, Illinois J.~Ma th. 7 (1963)\, 79--96.\n\n[3] Cai Heng Li and Yan Zhou Zhu\, A Proof of Gr oss' Conjecture on 2-Automorphic 2-Groups\,\nhttps://arxiv.org/abs/2312.16 416 (2024).\n\n[4] Cai Heng Li and Yan Zhou Zhu\,\nThe finite groups with three automorphism orbits\, https://arxiv.org/abs/2403.01725 (2024).\n LOCATION:https://researchseminars.org/talk/SiN/58/ END:VEVENT END:VCALENDAR