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

Given a finite permutation group G with domain Ω, we associate two subsets of natural numbers to G, namely I(G,Ω) and 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) is an interval of natural numbers, whereas M(G,Ω) may not necessarily form an interval. Moreover, for a given subset of natural numbers 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 and M(G,Ω)=X.