Nilpotent Residual of a Finite Group

Abstract

Let $F$ be a nilpotent group acted on by a group $H$ via automorphisms and let the group $G$ admit the semidirect product $FH$ as a group of automorphisms so that $C_G(F)=1$. We prove that the order of ${\gamma }_{\infty }(G)$, the rank of ${\gamma }_{\infty }(G)$ are bounded in terms of the orders of ${\gamma }_{\infty }(C_G(H))$ and $H$, the rank of ${\gamma }_{\infty }(C_G(H))$ and the order of $H$, respectively in cases where either $FH$ is a Frobenius group; $FH$ is a Frobenius-like group satisfying some certain conditions; or $FH=⟨\alpha ,\beta ⟩$ is a dihedral group generated by the involutions $\alpha$ and $\beta$ with $F=⟨\alpha \beta ⟩$ and $H=⟨\alpha ⟩$.