Another character theoretic formula for base size
Coen del Valle · Archiv der Mathematik · 2025
Abstract A base for a permutation group G acting on a set $$\Omega $$ Ω is a sequence $$\mathcal {B}$$ B of points of $$\Omega $$ Ω such that the pointwise stabiliser $$G_{\mathcal {B}}$$ G 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 $$\{1,2,\dots , n\}$$ { 1 , 2 , ⋯ , n } is obtained.