The base size of a primitive diagonal group. (Q372677)

A base for a permutation group \(G\) acting on a set \(\Omega\) is a subset of \(\Omega\) whose pointwise stabiliser in \(G\) is trivial; and the base size of \(G\) is the minimal size of such a base. Recent work by several authors aims to classify the primitive permutation groups with base size 2. In this paper, the author focuses on diagonal type groups.NEWLINENEWLINE In particular, the author proves that if the top group is not the alternating or the symmetric group (of degree the number of factors of the socle), then the base size is 2. Moreover, the author proves that, in that case, the proportion of pairs of \(\Omega\) that are bases tends to 1 as \(|G|\) grows to infinity.NEWLINENEWLINE In all other cases (that is, if the top group contains the alternating group), then the author determines the base size up to two possible values. A corollary of these results is that a conjecture of Pyber is verified for the case of diagonal groups: there is an absolute constant \(c\) such that the base size of a primitive permutation group \(G\) is at most \(c\log|G|/\log|\Omega|\). More precisely, for groups of diagonal type, the base size is bounded by \(\lceil\log|G|/\log|\Omega|\rceil+2\).NEWLINENEWLINE Finally, for a fixed number \(\geq 5\) of factors in the socle, the proportion of pairs of \(\Omega\) that are bases also tends to 1 as \(|G|\) grows to infinity.