Miscellaneous results on supersolvable groups

The paper contains two theorems generalizing the theorems of Huppert concerning the characterization of supersolvable and p -supersolvable groups, respectively. The first of these gives a new approach to prove Huppert's first named result. The second one has numerous applications in the paper. The notion of balanced pairs is introduced for non-conjugate maximal subgroups of a finite group. By means of them some new deep results are proved that ensure supersolvability of a finite group. Introduction We recall Huppert's characterizations for ( p -)supersolvable groups. (i) Let p be some prime. A finite group is p-supersolvable iff it is p-solvable and the index of any maximal subgroup is either p or coprime to p . (ii) A finite group is supersolvable iff all maximal subgroups of it have prime index . (See in [10, 9.2–9.5 Satz], pp. 717–718.) Among others it immediately follows that the class (formation) of finite supersolvable groups is saturated, i.e. the supersolvability of G /Φ( G ) is equivalent to the supersolvability of G itself. Result (ii) turned out to be of fundamental importance and it inspired a long series of further achievements. Concentrating to various characterizations of finite supersolvable groups by means of the index of maximal subgroups or the existence of cyclic supplements to maximal subgroups we mention [7], [12] and [15] from the past; cf. also [16] (or [6, Thm. 2.2], p 483).