On the Automorphism Groups of Quasiprimitive Almost Simple Graphs

✍ Scribed by Xin Gui Fang; George Havas; Cheryl E. Praeger

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 111 KB
• Volume: 222
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8020

✦ Synopsis

Let be a graph and let G be a subgroup of automorphisms of . Then G is said to be locally primitive on if, for each vertex v, the stabilizer G v induces a primitive group of permutations on the set of vertices adjacent to v. This paper investigates pairs G for which G is locally primitive on , G is an almost simple group (that is, L ≤ G ≤ Aut L for some nonabelian simple group L), and the simple socle L is transitive on vertices. Each such graph is a cover of a possibly smaller graph ˜ on which G is also locally primitive, and for which in addition Aut ˜ is quasiprimitive on vertices (that is, every nontrivial normal subgroup of Aut ˜ is vertex-transitive). It is proved that Aut ˜ is also an almost simple group. In the general case in which Aut is not quasiprimitive on vertices, we show that either every intransitive minimal normal subgroup of Aut centralizes L, or L is of Lie type and Aut involves an explicitly known same characteristic module for L.