Most primitive groups are full automorphism groups of edge-transitive hypergraphs
Group Theory
2014-09-09 v2
Abstract
We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. 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. Klin. Moreover, we give an upper bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph. This is essentially best possible.
Cite
@article{arxiv.1404.6739,
title = {Most primitive groups are full automorphism groups of edge-transitive hypergraphs},
author = {Laszlo Babai and Peter J. Cameron},
journal= {arXiv preprint arXiv:1404.6739},
year = {2014}
}
Comments
To appear in special issue of Journal of Algebra in memory of Akos Seress