On the generalized norms of finite groups

Abstract

The norm [Formula: see text] of a group [Formula: see text] is the intersection of the normalizers of all subgroups in [Formula: see text]. In this paper, the norm is generalized by studying on Sylow subgroups and [Formula: see text]-subgroups in finite groups which is denoted by [Formula: see text] and [Formula: see text], respectively. It is proved that the generalized norms [Formula: see text] and [Formula: see text] are all equal to the hypercenter of [Formula: see text].