On Sylow Normalizers of Finite Groups

Abstract

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

A survey on Sylow normalizers and classes of groups

“…• In general, n F = F, as shown by the above-referred examples and the last mentioned result in [17,Theorem 3.2].…”

“…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).…”

“…But also Examples 3.1 in [17] show that F = U = V is not enough to guarantee that n F = F (even if F ⊆ S or with Char(F) = P). In particular, the class of groups all whose {2, 3}-subgroups are nilpotent coincides with the class of groups with nilpotent Hall {2, 3}-subgroups.…”

“…In particular, the class of groups all whose {2, 3}-subgroups are nilpotent coincides with the class of groups with nilpotent Hall {2, 3}-subgroups. This is a covering formation F which satisfies the equivalent conditions in [17,Theorem 3.3], i.e. every critical group is either a Schmidt group or a cyclic group of prime order, but n F = F.…”

“…Then H is not a covering formation but n H = H. The class H × E π = (G | G = A × B, A ∈ H, B ∈ E π ) gives an example with full characteristic; see [16,Lemma 5.7]. Proposition 3.2 and Theorem 3.1 in [17] give conditions to guarantee that a subgroup-closed saturated formation H satisfying n H = H is a covering formation, in terms of the canonical local definition of H.…”