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

Abstract Given a finite permutation group G with domain Ω, we associate two subsets of natural numbers to G, namely {I} (G, ) I (G, Ω) and {M} (G, ) M (G, Ω), which are the sets of cardinalities of all the irredundant and minimal bases of G, respectively. We prove that {I} (G, ) I (G, Ω) is an interval of natural numbers, whereas {M} (G, ) 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, ) = X I (G, Ω) = X and {M} (G, ) = X M (G, Ω) = X.