Finite groups with a nilpotency condition

Abstract

Let [Formula: see text] and [Formula: see text] be positive integer numbers. In this paper, we study [Formula: see text], the class of all groups [Formula: see text] that for all subsets [Formula: see text] and [Formula: see text] of [Formula: see text] containing [Formula: see text] and [Formula: see text] elements, respectively, there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for [Formula: see text] and [Formula: see text] such that [Formula: see text] implies that [Formula: see text] is nilpotent. Also we will characterize all finite [Formula: see text]-groups in [Formula: see text], which [Formula: see text].