Abstract
Let [Formula: see text] be a hereditary saturated formation such that [Formula: see text] ([Formula: see text] denotes the class of all Sylow tower groups of type [Formula: see text]). In this paper, the concept of a fan of a subgroup of a group [Formula: see text], introduced in 1979 by Borevich, is used to prove that [Formula: see text] belongs to [Formula: see text]. We proved that a finite group [Formula: see text] if and only if [Formula: see text] and every basic subgroup of the fan of every Sylow subgroup in [Formula: see text] is [Formula: see text]-subnormal. We have also obtained that the group [Formula: see text] lies in [Formula: see text] if and only if all the basic subgroups of the fans of all Sylow subgroups in [Formula: see text] lie in [Formula: see text].
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