A characterization of the finite soluble groups

For G a group and M a subgroup of G, we say that a subgroup A of G is a supplement to M in G, if G = MA. We prove the conjecture of O.H. Kegel that a finite group whose maximal subgroups admit an abelian supplement is soluble. But this condition does not characterize the soluble groups among the finite groups. We prove that a finite group G is soluble if and only if every maximal subgroup M of G admits a supplement whose commutator subgroup is contained in M. Moreover, we determine the finite groups whose maximal subgroups have a nilpotent (resp. soluble) supplement. The latter groups still deserve a further analysis.