Collineation groups of projective planes of order n (Q1102536)

The paper is an attempt to make progress towards proving the conjecture that every projective plane has a nontrivial collineation. Since it is known that a projective plane of order 10 (if one exists) has a trivial collineation group, the conjecture would show that no such planes exist. Let X be a projective plane of order n, and \(x_ 0\in \ell_ 0\) a flag in X. Then the author proves that there is an \(n^ 3\)-dimensional vector space H(X) over GF(2), and the given flag equips H(X) with an alternating bilinear form which is nondegenerate when n is even. The group of those collineations of the affine plane \(X-\ell_ 0\) which fix the direction \(x_ 0\) is isomorphic to a double stabilizer of the isometry group of the alternating form. When n is even, this group is a double stabilizer of the symplectic group \(Sp(n^ 3,2)\). Some of the methods of the paper are graph-theoretic; a procedure is being introduced employing the symplectic group for the study of graphs. The author makes frequent use of his own previous results [J. Algebra 92, 128-149 (1985; Zbl 0562.51010); Discrete Math. 64, 43-79 (1987)].