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Let  G  be a finite group. We define the-normof  G  to be the intersection of the normalizers of the-residuals of all subgroups whose-residuals are nilpotent. In the present paper, we analyze the relationship between the-norm and-norm of finite groups, and the properties of the new normare investigated. Some new characterizations of finite groups with regard to the new definitions are given, and we also employ our results further to study the-norm of finite groups.

On the -norm and -norm of finite groups