Geodesic
On the intersections of solvable Hall subgroups in finite groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 74-83

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The following conjecture is considered: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. Under additional conditions on $G$ and $H$, it is shown that a minimal counterexample to this conjecture must be an almost simple group of Lie type.
Keywords: solvable Hall subgroup, finite simple group, $\pi$-radical. 
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     title = {On the intersections of solvable {Hall} subgroups in finite groups},
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E. P. Vdovin; V. I. Zenkov. On the intersections of solvable Hall subgroups in finite groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 74-83. http://geodesic.mathdoc.fr/item/TIMM_2009_15_2_a6/