A character theoretic formula for the base size

Abstract

A base for a permutation group G acting on a set \(\Omega \) is a sequence \({\mathcal {B}}\) of points of \(\Omega \) such that the pointwise stabiliser \(G_{{\mathcal {B}}}\) is trivial. The base size of G is the size of a smallest base for G. We derive a character theoretic formula for the base size of a class of groups admitting a certain kind of irreducible character. Moreover, we prove a formula for enumerating the non-equivalent bases for G of size \(l\in {\mathbb {N}}.\) As a consequence of our results, we present a very short, entirely algebraic proof of the formula of Mecenero and Spiga for the base size of the symmetric group \(\textrm{S}_n\) acting on the k-element subsets of \(\{1,2,3,\ldots ,n\}.\) Our methods also provide a formula for the base size of many product type permutation groups.