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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.
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v50n4.17.01}, url = {http://global-sci.org/intro/article_detail/jms/11319.html} }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.