We generalise the work of Segre (1965), Cameron -Goethals -Seidel (1978), and by showing that nontrivial m-ovoids of the dual polar spaces DQ(2d, q), DW(2d β 1, q) and DH(2d β 1, q 2 ) (d 3) are hemisystems. We also provide a more general result that holds for regular near polygons.2010 Mathematics Subject Classification. 05B25, 51E12, 51E20.
Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that PSL(2, π) Γ PSL(2, π) acting in its diagonal action on PSL(2, π) is separating, and hence synchronising, for π = 13 and π = 17. Furthermore, we show that such groups are non-spreading for all prime powers π.
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