International Congress of Mathematicians

ICM 2002 • Vol. II • 67^76Permutation Groups andNormal SubgroupsCheryl E. Praeger*AbstractVarious descending chains of subgroups of a finite permutation group canbe used to define a sequence of 'basic' permutation groups that are analoguesof composition factors for abstract finite groups. Primitive groups have beenthe traditional choice for this purpose, but some combinatorial applicationsrequire different kinds of basic groups, such as quasiprimitive groups, that aredefined by properties of their normal subgroups. Quasiprimitive groups admitsimilar analyses to primitive groups, share many of their properties, and havebeen used successfully, for example to study s-arc transitive graphs. Moreoverinvestigating them has led to new results about finite simple groups.2000 Mathematics Subject Classification: 20B05, 20B10 20B25, 05C25.Keywords and Phrases: Automorphism group, Simple group, Primitivepermutation group, Quasiprimitive permutation group, Arc-transitive graph.1. IntroductionFor a satisfactory understanding of finite groups it is important to study simplegroups and characteristically simple groups, and how to fit them together to formarbitrary finite groups. This paper discusses an analogous programme for studyingfinite permutation groups. By considering various descending subgroup chains offinite permutation groups we define in §2 sequences of 'basic' permutation groupsthat play the role for finite permutation groups that composition factors or chieffactors play for abstract finite groups. Primitive groups have been the traditionalchoice for basic permutation groups, but for some combinatorial applications largerfamilies of basic groups, such as quasiprimitive groups, are needed (see §3).Application of a theorem first stated independently in 1979 by M. E. O'Nanand L. L. Scott [4] has proved to be the most useful modern method for identifyingthe possible structures of finite primitive groups, and is now used routinely for their* Department of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway,Crawley, Western Australia 6009, Australia. E-mail: praeger@maths.uwa.edu.au