The base size of a primitive diagonal group
arXiv:1205.2079 · doi:10.1016/j.jalgebra.2012.11.020
Abstract
A base B for a finite permutation group G acting on a set X is a subset of X with the property that only the identity of G can fix every point of B. We prove that a primitive diagonal group G has a base of size 2 unless the top group of G is the alternating or symmetric group acting naturally, in which case a tight bound for the minimal base size of G is given. This bound also satisfies a well-known conjecture of Pyber. Moreover, we prove that if the top group of G does not contain the alternating group, then the proportion of pairs of points that are bases for G tends to 1 as |G| tends to infinity. A similar result for the case when the degree of the top group is fixed is given.
24 pages
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