On finite groups with weakly σ-normal subgroups

Let [Formula: see text] be a finite group and [Formula: see text]=[Formula: see text] some partition of the set of all primes. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text] [Formula: see text]-[Formula: see text] in [Formula: see text] if there exists a [Formula: see text]-subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is either [Formula: see text]-permutably embedded or [Formula: see text]-semipermutable in [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-normal subgroups on the structure of finite groups.