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

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order $(q-1,q+1)$ obtained by Payne derivation from the classical symplectic quadrangle $\mathsf{W}(3,q)$. For $q=p^f$ with $f\geq 2$ we obtain at least two nonisomorphic groups when $p\geq 5$ and at least three nonisomorphic groups when $p=2$ or $3$. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial $p$-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.Comment: some minor changes (including to title) after referee's comment

Point-primitive generalised hexagons and octagons

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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