On subplanes of Miquelian inversive planes (Q1021326)

An inversive plane is said to be Miquelian when the theorem of Miquel holds: if the circles \(c_0,\dots, c_3\) are such that \(c_{i\pmod 4}\cap c_{i+1\pmod 4}=\{a_i,b_i\}\), then \(a_0,\dots,a_3\) are concyclic if and only if \(b_0,\dots,b_3\) are concyclic. The classical example is the geometry consisting of the points on a sphere and the plane sections in the ordinary \(3\)-dimensional projective space. The main result of the paper is that every subplane of a Miquelian inversive plane of finite order occurs as a fixed point structure of a planar automorphism, i.e. an automorphism whose set of fixed points contains four non-concyclic points. It is also shown that two subplanes of the same order of a finite Miquelian inversive plane must coincide whenever they share more than two points. This is achieved by revisiting the well-known algebraic representation of Miquelian planes as given in [\textit{B. L. van der Waerden} and \textit{L. J. Smid}, Math. Ann. 110, 753--776 (1935; Zbl 0010.26803)], and more recently in [\textit{A. Lenard}, Abh. Math. Sem. Univ. Hamb. 65, 57--82 (1995; Zbl 0851.51007)] and in [\textit{H.-J. Kroll} and \textit{S. G. Taherian}, Abh. Math. Sem. Univ. Hamb. 69, 159--166 (1999; Zbl 0953.51002)], where it is proved that every Miquelian inversive plane is isomorphic to the chain geometry \(\Sigma(K:L)\) for some quadratic field extension \(L:K\). The following plane-subplane algebraic representation is a key lemma for the main results of the paper: if \(M'=(P',C')\) is a subplane of a Miquelian inversive plane \(M=(P,C)\) then there exist two quadratic field extension \(L:K\) and \(L':K'\) where \(L'\) (resp. \(K'\)) is a subfield of \(L\) (resp. \(K\)), and an isomorphism \(\phi:M\to \Sigma(K,L)\) such that \(\phi(M')=\Sigma(K',L')\).