×

zbMATH — the first resource for mathematics

Finite quasiprimitive graphs. (English) Zbl 0881.05055
Bailey, R. A. (ed.), Surveys in combinatorics, 1997. Proceedings of the 16th British combinatorial conference, London, UK, July 1997. London: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 241, 65-85 (1997).
The paper is a survey of results on quasiprimitive groups and finite arc-transitive graphs with vertex-quasiprimitive automorphism groups. The concept of a quasiprimitive group action is a generalization of the one of a primitive action, and a permutation group on a set \(\Omega\) is said to be quasiprimitive if all its non-trivial normal subgroups are transitive on \(\Omega\). The author successfully argues the importance of the study of quasiprimitive graphs for a better understanding of several classes of arc-transitive graphs; the claim is supported by a series of examples. An O’Nan-Scott type classification of quasiprimitive groups is presented and applied to the class of finite quasiprimitive 2-arc transitive vertex-transitive graphs. A discussion of the full automorphism groups of quasiprimitive graphs concludes the paper that includes an extensive list of open problems.
For the entire collection see [Zbl 0869.00029].

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
PDF BibTeX XML Cite