On the cardinality of irredundant and minimal bases of finite permutation groups

Abstract

Given a finite permutation group $G$ with domain $\Omega$, we associate two subsets of natural numbers to $G$, namely $\mathcal{I}(G,\Omega)$ and $\mathcal{M}(G,\Omega)$, which are the sets of cardinalities of all the irredundant and minimal bases of $G$, respectively. We prove that $\mathcal{I}(G)$ is an interval of natural numbers, whereas $\mathcal{M}(G,\Omega)$ may not necessarily form an interval. Moreover, for a given subset of natural numbers $X \subseteq \mathbb{N}$, we provide some conditions on $X$ that ensure the existence of both intransitive and transitive groups $G$ such that $\mathcal{I}(G,\Omega) = X$ and $\mathcal{M}(G,\Omega) = X$.