Geodesic
Existence of Hall normal subgroups in finite groups
Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 381-385

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We obtain a necessary and sufficient criterion for the existence of an invariant complement to a nilpotent subgroup contained as a direct factor in one of the maximal subgroups of a given group; we also find a condition for the $p$-closure of a group, all proper subgroups of which are $p$-closed, expressed in terms of the degree of one of its nonlinear irreducible characters. 
@article{MZM_1974_16_3_a3,
     author = {A. V. Romanovskii},
     title = {Existence of {Hall} normal subgroups in finite groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {381--385},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a3/}
}
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A. V. Romanovskii. Existence of Hall normal subgroups in finite groups. Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 381-385. http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a3/