A Conference in Celebration of Peter Cameron
نویسندگان
چکیده
s (updated 09 July 2013) Queen Mary, University of London, 8th – 10th July 2013 László Babai (University of Chicago, USA.) Symmetry Versus Regularity. The theme of this talk is the long-recognised seeming paradox that regularity constraints often impose severe limitations on the number and structure of symmetries, the archetype of such results being the fact that doubly transitive permutation groups, other than the alternating and the symmetric groups, are tiny: their orders are quasipolynomially bounded (less than exponential of a polynomial of the logarithm; in this case, a quadratic polynomial). In the talk, I will report brand new developments in an area that connects some of Peter Cameron’s favourite topics, including finite geometries, strongly regular graphs, primitive permutation groups, and his recent favourite, combinatorial relaxations to the notion of a base of a permutation group. The questions are partly motivated by a problem Peter posed a quarter-century ago in connection with his study of oligomorphic groups. The new results give a quasipolynomial bound on the number of automorphisms of Steiner t-designs and give new bounds and asymptotic structural constraints on the automorphism groups of strongly regular graphs. One of the results asserts that if X is a strongly regular graph with automorphism group G then, with known (trivial) exceptions, G has a subgroup of quasipolynomial index that is a Γμ group, i.e., a group of which every composition factor is a subgroup of the symmetric group of degree μ. (μ denotes the number of common neighbours of a pair of non-adjacent vertices
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