Projectivities, stabilizers and the Tutte group in projective planes (Q1320077)

Let \(\pi\) be a group of projectivities of a line \(L\ni\infty\) in a non- pappian projective plane \({\mathfrak P}(T)\) over a ternary field \(T\), and \(\pi_ \infty\) its stabilizer of \(\infty\). Let \(R_ a\) denote the extended radical of \(T\), namely the normal subloop of \(T^*\) generated by all \(r \overline{\in} T^*\) for which elements \(a,b,c,\dots\) exist such that at least one of the following equations holds: \(T(m,x,a)- T(m,x,b)= r(a-b)\), \(T(n,x,d)- T(m,x,c)= r((n-m) (x-y))\), \(x(yz)= r((xy)z)\), \(xy= r(yx)\). In connection with multiple-valued order functions it was shown [\textit{W. Junkers} and the author, J. Geom. 37, No. 1/2, 35-104 (1990; Zbl 0706.17001)] that for all \(\psi \overline{\in} \pi_ \infty\) and for all \(x \overline{\in} T\) one has \(\psi(x) \overline{\in} R_ a(T) \cdot x\cdot [\psi(1)- \psi(0)]+ \psi(0)\). Now, two-point and three-point stabilizer of \(\pi\) are investigated. Theorem: The mapping \(\psi: \pi_{0,\infty}\to T^*/Ra\), \(\phi\to R_ a \phi(1)\) is an epimorphism. Its kernel is the normal subgroup \(N\subset \pi\) generated by \(\pi_{0,1,\infty}\). From this one gets \(R_ a x= \{\phi(x)\); \(x\overline {\in} N\}\). Similarly, for \(\Delta:= \{x\to x+d\); \(d\overline{\in} T\}\) one has an epimorphism \(\pi_ \infty\to T^*/Ra\), \(\phi\to Ra [\phi(1)- \phi(0)]\) with kernel \(\Delta N\). The case \(T^*= Ra\) is equivalent with the nonexistence of multiple-valued halforderings and with the property that \(\pi_ q\) is generated by those projectivities which fix \(q\) and at least two further points. Let \(Q\) be the normal subloop of \(T^*\) generated by \(Ra\) and all products in which each factor occurs even time. Then \(T^*/Q\cong \pi/ \langle \pi_{0,1,\infty} \rangle\) and \(T^*=Q\) is equivalent with the nonexistence of Sperner halforderings and with the property that \(\pi\) is generated by its three-points stabilizers. Thus, for the author \(T^*/Q\) can serve as a measure for an inhomogenity of the plane that is not directly reflected by the Lenz-Barlotti-classification.