A class of graphs arising from the action of \(\text{PSL}(2,q^ 2)\) on cosets of \(\text{PGL}(2,q)\)
Motivated by the study of vertex-transitive graphs (those with a vertex- transitive automorphism group), this paper first gives a detailed description of the cosets of \(\text{PSL} (2,q)\) in \(\text{PSL}(2,q^ 2)\) (for any odd prime power \(q\)). That description is then used to describe the action of \(\text{PSL}(2,q)\) on these cosets and on the cosets of a subgroup isomorphic to \(\text{PGL}(2,q)\) which is maximal in \(\text{PSL}(2, q^ 2)\). Specifically the paper gives a detailed description of all suborbits (orbits of a point-stabilizer subgroup) of this primitive permutation action. The proofs use an interesting collection of elementary linear algebra calculations with \(2\times 2\) matrices over \(\text{GF}(q^ 2)\). (There seems to be a misprint in the top right entry of Table 1, which should say \(q(q + 1)\) instead of \(q(q - 1)\), and a few misprints in Lemma 4.1 (i).) The paper also briefly describes how each of these suborbits gives rise to an undirected vertex- transitive graph which is not a Cayley graph.
scientific article; zbMATH DE number 701686
scientific article