(untitled) (Journal of Mathematical Study, 2021)
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 :x,y is nilpotent}. G is called an N-group (n-group) if Nil_G(x) is a subgroup (nilpotent subgroup) of G for all x ∈ GZ^∗(G) where Z^∗(G) is the hypercenter of G. In the present paper, we determine finite N-groups in which the centraliser of each noncentral element is abelian. Also we classify all finite n-groups.