Saturated Formations Closed Under Sylow Normalizers

Finite groups in which Sylow normalizers have nilpotent Hall supplements

“…Hence, G has a nilpotent Hall {q, r}-subgroup and so we may assume that G r ⊆ N G (G q ). This is contrary to (3). Therefore |H 2 | | |N G (G s )| and hence N G (G s ) contains some conjugate of H 2 by Lemma 2.…”

“…W. Guo [2] proved that if the normalizer of each Sylow subgroup of a group G has a Hall complement in G then G is a dispersive group. In [3], the relationship of normalizers and formations was described. More recently, interest has centered on the indices of the normalizers of Sylow subgroups.…”

A survey on Sylow normalizers and classes of groups

“…In this direction relevant results appear in relation with subgroup lattices, factorized groups, formations and Fitting classes (see [1] for an account of this development). A further insight appears in [9,16,17] in relation with Sylow normalizers. Again a convenient restriction on the sets of primes locally defining a nilpotent-like Fitting formation leads to the so-called covering formations, which are classes of groups with nilpotent Hall subgroups for adequate sets of primes and play an important role in relation with Sylow normalizers (see Section 4).…”

“…In [7] conditions are found when some classes of soluble groups, defined by properties of the Sylow normalizers, are saturated formations. In particular, the class of soluble groups whose normalizers of Sylow p-subgroups are p-supersoluble, for a given prime p, is a saturated formation.…”

“…We emphasize at this point that classes of groups, being collections of groups closed under isomorphisms, provide a convenient framework for this investigation. D'Aniello, De Vivo and Giordano [6,7] led the study towards the theory of classes of groups and focused on closure properties of the class…”