Abstract
The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.
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