On a special class of Frobenius groups admitting planar partitions (Q2730492)

Let \(G\) be a group. A partition of \(G\) into subgroups is a collection \(\mathcal P\) of subgroups with pairwise trivial intersection such that \(G=\cup\mathcal P\). Finite groups with partition play their role in the classification of Zassenhaus transitive groups [\textit{M. Suzuki}, Arch. Math. 12, 241-254 (1961; Zbl 0107.25902)], see also Chapter 3 of \textit{R. Schmidt} [Subgroup lattices of groups. Berlin: Walter de Gruyter (1994; Zbl 0843.20003)]. NEWLINENEWLINENEWLINEEvery group \(G\) with partition \(\mathcal P\) yields an incidence structure \((G,\mathcal G)\), where \(\mathcal G:=\{gH\mid g\in G,H\in\mathcal P\}\) [cf. \textit{J. André}, Math. Z. 76, 85-102, 155-163, 240-256 (1961; Zbl 0099.15401)]. NEWLINENEWLINENEWLINEIf \(G\) is a Lie group with partition \(\mathcal P\) such that each element of \(\mathcal P\) is a closed subgroup then the collection \(\text{L}\mathcal P:=\{\text{L}H\mid H\in\mathcal P\}\) of Lie algebras forms a partition of the Lie algebra \(\text{L}G\). The partition \(\mathcal P\) is called planar if \(2\dim\text{L}H=\dim\text{L}G\) for each \(H\in\mathcal P\). The author has shown [Geom. Dedicata 83, 217-228 (2000; Zbl 0967.51004)] that, for a Lie group \(G\) with planar partition \(\mathcal P\), compactness of \(\text{L}\mathcal P\) in the Grassmann topology is equivalent to the fact that \((G,\mathcal G)\) forms a stable plane (i.e., intersection and joining are continuous operations with open domain). On the other hand, it is known that compactness of a planar partition of a real vector space is the crucial point for the construction of a compact connected translation plane [\textit{P. Breuning}, Mitt. Math. Sem. Gießen 86 (1970; Zbl 0201.53203), see also \textit{R. Löwen}, J. Geom. 36, 110-116 (1989; Zbl 0694.51011)]. (Locally) compact translation planes are well understood [cf. \textit{H. Salzmann} et al., Compact projective planes, Berlin (1995; Zbl 0851.51003)]. If a compact planar partition is invariant under conjugation, the corresponding stable plane also carries the structure of a symmetric space in such a way that the symmetries are automorphisms of the incidence structure [\textit{H. Löwe}, Math. Z. 232, No. 2, 197-216 (1999; Zbl 0942.51008)]. NEWLINENEWLINENEWLINEIt remains to find Lie groups with planar partitions. \textit{P. Plaumann} and \textit{K. Strambach} [Forum Math. 2, 523-578 (1990; Zbl 0722.22005); Semin. Sophus Lie 1, 103-106 (1991; Zbl 0757.22002)] have shown that for such a group either the exponential map is a diffeomorphism from \(\text{L}G\) onto \(G\), or \(G\) is a special type of Frobenius group: there is a representation of \(C\in\{{\mathbb C}^\times,{\mathbb H}^\times\}\) by fixed-point free linear bijections of a real vector space \(V\) such that \(G\) is the corresponding semidirect product of \(C\) by \(V\). The author shows that the existence of a planar partition on such a Frobenius group with \(C={\mathbb C}^\times\) implies that \(V\) is a complex vector space, and \(C\) acts as usual. For \(C={\mathbb H}^\times\) there are two cases: either \(V\) is a quaternion vector space, and \(C\) acts as usual, or \(\dim_{\mathbb R}V=4\), and \(C\) induces the multiplication of one of the Kalscheuer nearfields [\textit{F. Kalscheuer}, Abh. Math. Sem. Hansischen Univ. 13, 413-435 (1940; Zbl 0023.00602); cf. \textit{J. Tits}, Comment. Math. Helv. 26, 203-224 (1952; Zbl 0047.26002); 30, 234-240 (1956; Zbl 0070.02505)]. In the last case, there is only one planar partition. In all other cases, there are different possibilities. However, the study of these partitions is completely reduced to the study of planar partitions of the Lie algebras.