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

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