Locally primitive Cayley graphs of finite simple groups. (English) Zbl 1015.05034
X. G. Fang and C. E. Praeger [J. Algebra 205, 37-52 (1998; Zbl 0913.05051)] and X. G. Fang, G. Havas, C. E. Praeger [J. Algebra 222, 271-283 (1999; Zbl 0980.05032)] started to investigate the finite connected nonbipartite graphs \(\Gamma\) admitting an almost simple group \(G\) of automorphisms such that \(G\) is transitive on the arcs of \(\Gamma\), \(G\) is locally primitive on \(\Gamma\), and \(G\) is quasiprimitive on the vertices of \(\Gamma\). The aim was to deduce, from these properties of the group \(G\), certain properties of the full automorphism group \(\operatorname{Aut}(\Gamma)\) of \(\Gamma\). In these papers it has been assumed that the simple socle \(L\) of \(G\) does not act semiregulary on the vertices of \(\Gamma\). In the paper under review the authors assume that \(L\) is semiregular, hence regular, on the vertices of \(\Gamma\), where \(G, L\) and \(\Gamma\) are as above. They show that \(L\) is a simple group of Lie type and that \(\operatorname{Aut}(\Gamma)\) is a subgroup of \(\operatorname{Aut}(L)\).
Reviewer: B.Baumeister (Halle)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 05C75 | Structural characterization of families of graphs |
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