Finite $2$-groups with exactly three automorphism orbits

TL;DR

This work classifies finite $3$-orbit $2$-groups, i.e., groups $G$ for which $\operatorname{Aut}(G)$ acting on $G$ has exactly three orbits. By translating the problem into a squaring data framework encoded by a consistent power-commutator presentation, the authors show the automorphism action is solvable and reduce the classification to Dornhoff’s list of Suzuki $2$-groups. They then identify the full set of corresponding groups as the Suzuki $2$-groups of three families: $A(n,\theta)$, $B(n,\operatorname{id},\mu+\mu^{-1})$, and the exceptional $B(3,\theta,\varepsilon)$, with explicit isomorphisms such as $P(\varepsilon)\cong B(3,\theta,\varepsilon)$. The results connect a structural, combinatorial encoding (the squaring map) with classical Suzuki–Dornhoff theory, completing the finite $3$-orbit $2$-group classification and clarifying relationships among the known examples.

Abstract

We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All of them are Suzuki $2$-groups, and they appear in an earlier classification of Dornhoff.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Proposition 3.1
• proof