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1、AssociationschemesandpermutationgroupsCombinatoricsStudyGroupNotesbyPeterCameronAbstractIntheStudyGroup,wediscussedtheO'Nan{ScottTheoremandconsidereditsapplicationtotheproblem:Whichtransitivepermuta-tiongroupshavethepropertythattheypreserveonlythetrivialass
2、o-ciationscheme?Weshowthatsuchagroupisprimitive,andthatitiseitherofdiagonaltypeoralmostsimple.Moreover,inthediagonalcase,thesoclehasatleastfoursimplefactors.Itisstillopenwhetherdiagonalgroupswiththispropertyexist,buttherearealmostsimpleexamples.1Transitivit
3、y,primitivity,andO'Nan{ScottInthissection,wegobrie
ythroughsometopicswhicharecoveredinmoredetailin[2,3].Letbea niteset.Oftenwetake=f1;:::;ng.ThesymmetricgroupSym()isthegroupofallpermutationson,withtheoperationofcomposition.Wewritepermutationsontheright,soth
4、attheimageofunderthepermutationgiswritten g;thenghdenotesthecompositiong rst,thenh.If=f1;:::;ng,wewriteSforSym().nApermutationgrouponisasubgroupofSym().Moregenerally,anactionofGonisahomomorphismfromGtoSym()(sothattheimageofanactionisapermutationgroup).Anyde
5、 nitiongivenbelowforpermutationgroupscanbegeneralisedtoactions,byrequiringittoholdfortheimage.TwoactionsofG,onand,aresaidtobeisomorphicifthereisa12bijection :!suchthat,forany2andg2G,wehave121( g) =( )g;1wheretheactionsofgontheleftandrightarethoseonandrespe
6、ct-12ively;inotherwords,eachelementofGinducesthesamepermutationonthetwosets,uptore-namingtheirelements.LetGbeapermutationgroupon.For2,theorbitofunderG,Gdenote,isthesetofallelementsofwhichcanbeobtainedbyapplyingelementsofGto:G=f g:g2Gg:Ifthereisjustoneorbit,
7、(thatis,if,forany ;2thereexistsg2Gwith g=),thenwesaythatGistransitive.ThestabiliserofisthesubgroupofGconsistingofallelements xing:G=fg2G: g=g:Now,ifGistransitive,thenthestabilisersareallconjugate.Forasimple?1calculationshowsthat,if g=,thengGg=G.Anexampleofa
8、transitiveactionofGisthatontheset(G:H)ofrightcosetsofasubgroupH,wheretheactionisbyrightmultiplication:theelementg2GinducesthepermutationHx7!H(xg).Uptoisomorphism,everytransitiveactionisofthisform:Theor
