A pseudo-hyperoval of a projective space PG(3n − 1, q), q even, is a set of q n + 2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudohyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point-primitive, linetransitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that Q is flag-transitive and isomorphic to T * 2 (H), where H is either the regular hyperoval of PG(2, 4) or the Lunelli-Sce hyperoval of PG(2, 16). (i) H acts transitively on the lines of Q;(ii) H acts transitively on the flags of Q; (iii) H acts transitively on the pseudo-hyperoval {U 1 , U 2 , . . . , U t+1 }, by conjugation.Theorem 1.1 is proved in Section 2. It implies that a classification of transitive pseudohyperovals would yield a classification of the generalized quadrangles that admit a line-transitive automorphism group with a point-regular abelian normal subgroup, and, moreover, that such generalized quadrangles are, in fact, flag-transitive. By a result of J. A. Thas [28, §4.5], if a projective space PG(3n − 1, q) contains a pseudo-hyperoval, then q = 2 f for some positive integer f . For small values of the product nf , we appeal to some existing results to classify the transitive pseudo-hyperovals of PG(3n − 1, 2 f ), Journal of Combinatorial Designs
The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group G preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on G, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that G cannot have holomorph compound O'Nan-Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in non-Abelian finite simple groups, and about fixities of primitive permutation groups.2010 Mathematics Subject Classification. primary 51E12; secondary 20B15, 05B25.
Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set P of Γ, and transitively on the lines of Γ. We show that the primitive action of G on P cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on P is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.2010 Mathematics Subject Classification. primary 51E12; secondary 20B15, 05B25.Section 5] for primitive permutation groups, this means that the action of G on points is of either holomorph simple (HS) or holomorph compound (HC) type (see Section 2 for definitions). We prove the following result.Theorem 1.1. Let G be a collineation group of a finite thick generalised quadrangle with point set P and line set L. If G acts transitively on L and primitively on P, then G has a unique minimal normal subgroup; that is, the action of G on P does not have O'Nan-Scott type HS or HC.
For all n > 2, we study nth order generalisations of Riemannian cubics, which are second-order variational curves used for interpolation in semiRiemannian manifolds M. After finding two scalar constants of motion, one for all M, the other when M is locally symmetric, we take M to be a Lie group G with bi-invariant semi-Riemannian metric. The Euler-Lagrange equation is reduced to a system consisting of a linking equation and an equation in the Lie algebra. A Lax pair form of the second equation is found, as is an additional vector constant of motion, and a duality theory, based on the invariance of the Euler-Lagrange equation under group inversion, is developed. When G is semisimple, these results allow the linking equation to be solved by quadrature using methods of two recent papers; the solution is presented in the case of the rotation group SO(3), which is important in rigid body motion planning.
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