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 sup plement 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 solubl e. But this condition does not characterize the soluble groups among the fi nite 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 c ontained in M. Moreover, we determine the finite groups whose maximal subgr oups have a nilpotent (resp. soluble) supplement. The latter groups still d eserve a further analysis.