Geodesic
On intersections of nilpotent subgroups in finite symmetric and alternating groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 144-149

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It is proved that, in a nonsolvable finite symmetric or alternating group, for any pair of nilpotent subgroups, there exists a subgroup conjugate to one of them such that its intersection with the other subgroup is trivial, except for the group $S_8$.
Keywords: maximal nilpotent subgroup, symmetric group, alternating group. 
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     author = {V. I. Zenkov},
     title = {On intersections of nilpotent subgroups in finite symmetric and alternating groups},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {144--149},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a13/}
}
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V. I. Zenkov. On intersections of nilpotent subgroups in finite symmetric and alternating groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 144-149. http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a13/