Independence of axioms for fourgonal families
According to \textit{L. Bader} and \textit{S. E. Payne} [J. Geom. 63, No. 1-2, 1-16 (1998; Zbl 0934.51003)], four axioms characterize the elation groups \(E\) of elation generalized quadrangles: \(E\) contains a family \(S\) of subgroups and, for each \(A\in S\), another subgroup \(A^*\) containing \(A\), subject to the conditions (1) \(A^*B=E\), (2) \(A^*\cap B=1\), (3) \(AB\cap C=1\), (4) \(E=S^*\cup \bigcup_{B\in S}BA\); here, \(A,B,C\in S\) are arbitrary distinct elements. Moreover, these data (called a fourgonal family) suffice to construct the quadrangle. Conditions (2) and (3) are the original ones used by \textit{W. M. Kantor} in the finite case [Math. Z. 192, 45-50 (1986; Zbl 0592.51003)]. The authors show that none of the four conditions is a consequence of the other three. In order to violate condition (4), it suffices to omit one of the pairs \((A,A^*)\), but the authors give a more interesting example, which defines a geometry of diameter 6 like a real fourgonal family. The second author came across this example when he classified compact three-dimensional elation quadrangles [\textit{M. Stroppel}, Geom. Dedicata 83, No. 1-3, 149-167 (2000; Zbl 0986.51008)]. The counterexample to condition (1) is constructed by a transfinite process, and the one to condition (3) has an elementary abelian group \(E\) of order \(2^5\).
scientific article; zbMATH DE number 1760624