Association schemes and permutation groups Combinatorics Study Group Notes by Peter Cameron.pdfVIP

Asso ciation schemes and p ermutation groups Combinatorics Study Group Notes by Peter Cameron Abstract In the Study Group we discussed the ONanScott Theorem and considered its application to the problem Which transitive permuta tion groups have the pr operty that they pr eserve only the trivial asso ciation scheme We show that such a group is primitive and that it is either of diagonal typ e or almost simple Moreover in the diagonal case the so cle has at least four simple factors It is still op en whether diagonal groups with this prop erty exist but there are almost simple examples Transitivity primitivity and ONanScott In this section we go briey through some topics which are covered in more detail in Let b e a nite set Often we take f ng The symmetric gr oup Sym is the group of all p ermutations on with the op eration of comp osition We write p ermutations on the right so that the image of under the p ermutation g is written g then g h denotes the comp osition g rst then h If f ng we write S for Sym n A p ermutation gr oup on is a subgroup of Sym More generally an action of G on is a homomorphism from G to Sym so that the image of an action is a p ermutation group Any denition given b elow for p ermutation groups can b e generalised to actions by requiring it to hold for the image Two actions of G on 1 and 2 are said to b e isomorphic if there is a bijection such that for any and g G we have 1 2 1 g g where the acti