Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon \cS, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group.Comment: forgot to upload the appendices in version 1, and this is rectified in version 2. erased cross-ref keys in version 3. Minor revision in version 4 to implement the suggestion by the referee (new section at the end, extended acknowledgment, simpler proof for Lemma 4.2

Point-primitive generalised hexagons and octagons

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On collineations and dualities of finite generalized polygons

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation theta of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by theta. As a special case we consider generalized 2n-gons of order (1,t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t

Generalised quadrangles with a group of automorphisms acting primitively on points and lines

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.Comment: 20 page

Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 hep-tagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C73 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C42 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table