Vogler, Hans; Wresnik, Helmut On a theorem of Frobenius in the \(n\)-dimensional isotropic space \(I^k_n\). (Über einen Satz von Frobenius im \(n\)-dimensionalen isotropen Raum \(I^k_n\).) (German) Zbl 1157.51014 Grazer Math. Ber. 352, 118-122 (2008). Denote by \(H\) the subgroup of the isotropic affine group, described by block-diagonal matrices with two diagonal elements: An orthogonal matrix \(\mathbf A\) of dimension \((n-k,n-k)\) and a regular matrix \(\mathbf C\) of dimension \((k,k)\). The authors prove the following result: If \(G\) is a finite subgroup of the affine group in isotropic spaces such that there exists an isotropic vector \(\mathbf x\) whose orbit \(\{\lambda({\mathbf x})\mid\lambda\in G\}\) has maximal Euclidean dimension then \(G\) is conjugate to a finite subgroup of \(H\). For \(k = 0\) this is just the Theorem of Frobenius for the \(n\)-dimensional Euclidean space. In contrast to the Euclidean case, the conjugate subgroup \(\alpha\circ G\circ\alpha^{-1}\) is not necessarily contained in the isotropic motion group (the subgroup where \(\mathbf C\) is a lower triangular matrix). The notions and definitions used in the reviewed article (and in this review) can be found in H. Vogler and H. Wresnik [Grazer Math. Ber. 307, 46 p. (1989; Zbl 0682.53012)]. Reviewer: Hans-Peter Schröcker (Innsbruck) MSC: 51N25 Analytic geometry with other transformation groups 53A17 Differential geometric aspects in kinematics 53A35 Non-Euclidean differential geometry Keywords:isotropic geometry; Theorem of Frobenius Citations:Zbl 0682.53012 PDFBibTeX XMLCite \textit{H. Vogler} and \textit{H. Wresnik}, Grazer Math. Ber. 352, 118--122 (2008; Zbl 1157.51014)