A characterization of finite supersolvable groups. (Q2915405)

The class of supersolvable groups is an important class of finite groups lying between nilpotent and solvable groups. Many characterizations of these groups are known, maybe the most famous is Huppert's Theorem [see Hauptsatz 9.5 in \textit{B. Huppert}, Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0217.07201)]. For solvable groups, this theorem was generalized by \textit{O.-U. Kramer} [in Math. Z. 148, 89-97 (1976; Zbl 0311.20006)], saying that a finite solvable group is supersolvable if and only if \(|G:M|\) is prime for every maximal subgroup \(M\) of \(G\) containing the Fitting subgroup \(F(G)\).NEWLINENEWLINE The paper under review gives new characterizations of supersolvable groups by using the subgroup \(\widetilde F(G)\) introduced by \textit{L. A. Shemetkov} [Formations of finite groups. (Russian), Sovremennaya Algebra. Moskva: Nauka (1978; Zbl 0496.20014)]. Here, \(\widetilde F(G)\) satisfies \(\widetilde F(G)/\Phi(G)=\text{Soc}(G/\Phi(G))\). For example, it is proved that \(G\) is supersolvable if and only if \(|\widetilde F(G):\widetilde F(G)\cap M|\) is either 1 or prime for every maximal subgroup \(M\) of \(G\). Using this theorem, a common generalization of both Huppert and Kramer's characterizations is given.NEWLINENEWLINE As an application, the authors give a sufficient condition for a group \(G\) being supersolvable in terms of the Sylow subgroups of \(\widetilde F(G)\) by using the concept of weakly \(s\)-supplemented subgroups introduced by \textit{A. N. Skiba} [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)].