Geodesic
The nilpotency criterion for the derived subgroup of a finite group
Problemy fiziki, matematiki i tehniki, no. 3 (2017), pp. 58-60

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It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\geqslant |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.
Keywords: finite group, commutator, derived subgroup, nilpotent subgroup. 
@article{PFMT_2017_3_a9,
     author = {V. S. Monakhov},
     title = {The nilpotency criterion for the derived subgroup of a finite group},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {58--60},
     publisher = {mathdoc},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2017_3_a9/}
}
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V. S. Monakhov. The nilpotency criterion for the derived subgroup of a finite group. Problemy fiziki, matematiki i tehniki, no. 3 (2017), pp. 58-60. http://geodesic.mathdoc.fr/item/PFMT_2017_3_a9/