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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].Letbeaniteset.Oftenwetake=f1;:::;ng.ThesymmetricgroupSym()isthegroupofallpermutationson,withtheoperationofcomposition.Wewritepermutationsontheright,soth
4、attheimageofunderthepermutationgiswritteng;thenghdenotesthecompositiongrst,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=fg:g2Gg:Ifthereisjustoneorbit,
7、(thatis,if,forany;2thereexistsg2Gwithg=),thenwesaythatGistransitive.ThestabiliserofisthesubgroupofGconsistingofallelementsxing:G=fg2G:g=g:Now,ifGistransitive,thenthestabilisersareallconjugate.Forasimple?1calculationshowsthat,ifg=,thengGg=G.Anexampleofa
8、transitiveactionofGisthatontheset(G:H)ofrightcosetsofasubgroupH,wheretheactionisbyrightmultiplication:theelementg2GinducesthepermutationHx7!H(xg).Uptoisomorphism,everytransitiveactionisofthisform:Theor