A character theoretic formula for the 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. 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 $$l\in {\mathbb {N}}.$$ l ∈ 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 $$\textrm{S}_n$$ S n acting on the k-element subsets of $$\{1,2,3,\ldots ,n\}.$$ { 1 , 2 , 3 , … , n } . Our methods also provide a formula for the base size of many product type permutation groups.