Maximal subgroups of finite groups avoiding the elements of a generating set

We give an elementary proof of the following remark: if G is a finite group and { g1, … , gd} is a generating set of G of smallest cardinality, then there exists a maximal subgroup M of G such that M∩ { g1, … , gd} = ∅. This result leads us to investigate the freedom that one has in the choice of the maximal subgroup M of G. We obtain information in this direction in the case when G is soluble, describing for example the structure of G when there is a unique choice for M. When G is a primitive permutation group one can ask whether is it possible to choose in the role of M a point-stabilizer. We give a positive answer when G is a 3-generated primitive permutation group but we leave open the following question: does there exist a (soluble) primitive permutation group G= ⟨ g1, … , gd⟩ with d(G) = dCloseSPigtSPi 3 and with ⋂ 1 ≤ i ≤ dsupp (gi) = ∅? We obtain a weaker result in this direction: if G= ⟨ g1, … , gd⟩ with d(G) = d, then supp (gi) ∩ supp (gj) ≠ ∅ for all i, j∈ { 1 , … , d}.