A sufficient condition for nilpotency of the nilpotent residual of a finite group

Abstract

Abstract Let G be a finite group with the property that if a , b {a,b} are powers of δ 1 * {\delta_{1}^{*}} -commutators such that ( | a | , | b | ) = 1 {(|a|,|b|)=1} , then | a ⁢ b | = | a | ⁢ | b | {|ab|=|a||b|} . We show that γ ∞ ⁢ ( G ) {\gamma_{\infty}(G)} is nilpotent.