On the orbital diameter of classical groups in standard actions

Abstract Let G be a primitive permutation group acting on a finite set X . The orbital diameter $$\textrm{diam}(X,G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>diam</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is defined to be the supremum of the diameters of the (connected) orbital graphs of G after disregarding the directions of all edges in the graphs. This invariant is studied in the case when G is an almost simple group in a standard action. A lower bound is given for $$\textrm{diam}(X,G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>diam</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and we provide a partial classification of pairs ( X , G ) for which the orbital diameter is at most 2.