The flag-transitive \(C_ 3\)-geometries of finite order
A \(C_3\)-geometry \({\mathcal G}\) of finite order \((x,y)\) is a residually connected incidence geometry on \(I= \{0,1,2\}\), in which the residue of an element of type \(i\) is isomorphic to a generalized quadrangle of finite order \((x,y)\), to a generalized digon, or to a projective plane of order \(x\), respectively for \(i=0\), 1 or 2. It has been conjectured that a \(C_3\)-geometry of finite order \((x,y)\) with \(x\geq 2\) is either a finite building of type \(C_3\) or the sporadic \(A_7\)-geometry, if it admits a flag-transitive automorphism group. Several authors have contributed to the solution of special cases and related results, and the present paper finally offers a complete proof of the general conjecture. The proof relies on the classification of finite simple groups to the extent that it relies on a theorem of \textit{W. M. Kantor} [J. Algebra 106, 15-45 (1987; Zbl 0606.20003)] about flag-transitive non-desarguesian projective planes that in turn depends on the classification theorem.
scientific article; zbMATH DE number 894839
scientific article