Another character theoretic formula for 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. Extending the results of a recent paper of the author, we prove a 2013 conjecture of Fritzsche, Külshammer, and Reiche. Moreover, we generalise this conjecture and derive an alternative character theoretic formula for the base size of a certain class of permutation groups. As a consequence of our work, a third formula for the base size of the symmetric group of degree n acting on the subsets of \(\{1,2,\dots , n\}\) is obtained.