On Finite Groups Whose Nilpotentisers Are Nilpotent Subgroups

Abstract

Let $G$ be a finite group and $x ∈ G.$ The nilpotentiser of $x$ in $G$ is defined to be the subset $Nil_G(x) =\{y∈ G :\langle x,y \rangle \ is\ nilpotent\}.$ $G$ is called an $\mathcal{N}$-group ($n$-group) if $Nil_G(x)$ is a subgroup (nilpotent subgroup) of $G$ for all $x ∈ G\setminus Z^∗(G)$ where $Z^∗(G)$ is the hypercenter of $G$. In the present paper, we determine finite $\mathcal{N}$-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite $n$-groups.