Most primitive groups are full automorphism groups of edge-transitive hypergraphs

Babai, László and Cameron, P. J. (2015) Most primitive groups are full automorphism groups of edge-transitive hypergraphs. Journal of Algebra, 421. pp. 512-523. ISSN 0021-8693

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Abstract

We prove that, for a primitive permutation group G acting on a set X of size n, other than the alternating group, the probability that Aut(X, YG)=G for a random subset Y of X, tends to 1 as n→∞. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M.H. Klin. Moreover, we give an upper bound n1/2+ε for the minimum size of the edges in such a hypergraph. This is essentially best possible.

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