A Metanilpotency Criterion for a Finite Solvable Group

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.