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

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