On finite d-maximal groups

Let (Formula presented.) be a positive integer. A finite group is called (Formula presented.) -maximal if it can be generated by precisely (Formula presented.) elements, whereas its proper subgroups have smaller generating sets. For (Formula presented.), the (Formula presented.) -maximal groups have been classified up to isomorphism and only partial results have been proved for larger (Formula presented.). In this work, we prove that a (Formula presented.) -maximal group is supersolvable and we give a characterisation of (Formula presented.) -maximality in terms of so-called maximal (Formula presented.) -pairs. Moreover, we classify the maximal (Formula presented.) -pairs of small rank obtaining, as a consequence, the classification of the isomorphism classes of 3-maximal finite groups.