The base size of a primitive diagonal group
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 wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.2079
 Bibcode:
 2012arXiv1205.2079F
 Keywords:

 Mathematics  Group Theory;
 20B15
 EPrint:
 24 pages