Abstract
Let G be a finite group and let We prove that the coprime subgroup
is nilpotent if and only if
for any
-commutators
of coprime orders (Theorem A). Moreover, we show that the coprime subgroup
is nilpotent if and only if
for any powers of
-commutators
of coprime orders (Theorem B).
Acknowledgment
The authors wish to thank Professor Pavel Shumyatsky for interesting discussions and the anonymous referee for the insightful comments. Moreover, this study was carried out during the second author’s visit to the University of Brasilia. He wishes to thank the Department of Mathematics for the excellent hospitality.