Abstract
Denote by |x| the order of an element x of a group. An element of a group is called primary if its order is a nonnegative integer power of a prime. If a and b are primary elements of coprime orders of a group, then the commutator a−1b−1ab is called a *-commutator. The intersection of all normal subgroups of a group such that the quotient groups by them are nilpotent is called the nilpotent residual of the group. It is established that the nilpotent residual of a finite group is generated by commutators of primary elements of coprime orders. It is proved that the nilpotent residual of a finite solvable group is nilpotent if and only if |ab| ≥ |a||b| for any *-commutators of a and b of coprime orders.
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