Geodesic
Criterion for $\pi$-supersolvability for finite groups
Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 57-61 
N. M. Kurnosenko. Criterion for $\pi$-supersolvability for finite groups. Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 57-61. http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a8/
@article{MZM_1992_52_1_a8,
     author = {N. M. Kurnosenko},
     title = {Criterion for $\pi$-supersolvability for finite groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {57--61},
     year = {1992},
     volume = {52},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_1_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that the class of finite $\pi$-supersolvable groups is precisely the class of all finite $\pi$-solvable groups with the following property: For each maximal subgroup $M$ of a $\pi$-solvable group $G$ with index $p^{\alpha}$ for some $p\in\pi$, there exists a cyclic subgroup $S$ of order $p^{\beta}(\beta\geqslant\alpha)$ such that $G=MS$ and $S$ commutes with each element of the Sylow system $\Sigma_M$ of the subgroup $M$.