On one generalization of finite nilpotent groups

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Let σ={σᵢ | i∈ I} be a partition of the set P of all primes and G a finite group. A chief factor H/K of G is said to be σ-central if the semidirect product (H/K) (G/CG(H/K)) is a σᵢ-group for some i=i(H/K). G is called σ-nilpotent if every chief factor of G is σ-central. We say that G is semi-σ-nilpotent (respectively weakly semi-σ-nilpotent) if the normalizer NG(A) of every non-normal (respectively every non-subnormal) σ-nilpotent subgroup A of G is σ-nilpotent. In this paper we determine the structure of finite semi-σ-nilpotent and weakly semi-σ-nilpotent groups.