Finite quasiprimitive graphs. (English) Zbl 0881.05055

Praeger, Cheryl E.

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].

Reviewer: R.Jajcay (Terre Haute)

MSC:

05C25

Graphs and abstract algebra (groups, rings, fields, etc.)

Keywords:

quasiprimitive groups; automorphism groups; quasiprimitive graphs; arc-transitive graphs; vertex-transitive graphs; open problems