Finite Minimal Simple Groups Non-satisfying the Basis Property

Abstract

Let G be a finite group. We say that G has the Basis Property if every subgroup H of G has a minimal generating set (basis), and any two bases of H have the same cardinality. A group G is called minimal not satisfying the Basis Property if it does not satisfy the Basis Property, but all its proper subgroups satisfy the Basis Property. We prove that the following groups PSL(2, 5) ∼A5, PSL(2, 8) , are minimal groups non satisfying the Basis Property, but the groups PSL(2, 9), PSL(2, 17) and PSL(3, 4) are not minimal and not satisfying the Basis Property.