On Graphs Admitting Arc-Transitive Actions of Almost Simple Groups

✍ Scribed by Xin Gui Fang; Cheryl E Praeger

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 217 KB
• Volume: 205
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7383

✦ Synopsis

Let ⌫ be a finite connected regular graph with vertex set V ⌫, and let G be a subgroup of its automorphism group Aut ⌫. Then ⌫ is said to be G-locally primiti¨e if, for each vertex ␣ , the stabilizer G is primitive on the set of vertices adjacent to ␣ ␣. In this paper we assume that G is an almost simple group with socle soc G s S; that is, S is a nonabelian simple group and S eG F Aut S. We study nonbipartite ᎏ graphs ⌫ which are G-locally primitive, such that S has trivial centralizer in Aut ⌫ Ž . and S is not semiregular on vertices. We prove that one of the following holds: i

with Y almost simple and soc Y / S, or ᎏ Ž . iii S belongs to a very restricted family of Lie type simple groups of characteristic p, say, and Aut ⌫ contains the semidirect product Z d :G, where Z d is a known p p absolutely irreducible G-module. Moreover, in certain circumstances we can guar-Ž . Ž . antee that S eAut ⌫ F Aut S . For example, if ⌫ is a connected G, 2 -arc ᎏ Ž . Ž Ž .. Ž 2 nq1

. Ž. transitive graph with Sz q F G F Aut Sz q qs2 G8 or Gs Ree q Ž 2 nq1

. Ž . q s 3 G 27 , then G F Aut ⌫ F Aut G .