M.D. Pérez-Ramos

Fitting classes and products of totally permutable groups

The second and third authors have been supported by Proyecto PB 97-0674-C02-02 of DGESIC, Ministerio de Educacion y Cultura, Spain.

Injectors and Radicals in Products of Totally Permutable Groups

Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.

SATURATED FORMATIONS CLOSED UNDER SYLOW NORMALIZERS

In this article we show that a finite soluble group possesses nilpotent Hall subgroups for well-defined sets of primes if and only if its Sylow normalizers satisfy the same property. In fact, this property of groups provides a characterization of the subgroup-closed saturated formations, whose elements are characterized by the Sylow normalizers belonging to the class, in the universe of all finite soluble groups.

On Sylow normalizers of finite groups

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.

On the product of two π-decomposable soluble groups

Let the group G = AB be a product of two π-decomposable sub-groups A = Oπ(A) × Oπ′ (A) and B = Oπ(B) × Oπ′ (B) where π is a set of primes. The authors conjecture that Oπ(A)Oπ(B) = Oπ(B)Oπ(A) if π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if Oπ ′(A) and Oπ′(B) have coprime orders.

Persistent characterizations of injectors in finite solvable groups

Abstract In response to a question of Doerk and Hawkes [Finite soluble groups, de Gruyter, 1992, p. 553], we shall obtain characterizations of the injectors of a finite solvable group (without recourse to the concept of a Fitting set), and we also answer in the negative a question in [Dark and Feldman, J. Group Theory 9: 2006, p. 785].

On 2-generated subgroups and products of groups

Abstract For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.

A self-centralizing characteristic subgroup

In this note we introduce a self-centralizing characteristic subgroup, associated with quasinilpotent injectors, of a finite group.

A FAMILY OF DOMINANT FITTING CLASSES OF FINITE SOLUBLE GROUPS

In this paper a large family of dominant Fitting classes of finite soluble groups and the description of the corresponding injectors are obtained. Classical constructions of nilpotent and Lockett injectors as well as p-nilpotent injectors arise as particular cases.

Permutability in finite soluble groups

Let G be a finite soluble group and let Σ be a Hall system of G . A subgroup U of G is said to be Σ- permutable if U permutes with every member of Σ. In [1; I, 4·29] it is proved that if U and V are Σ-permutable subgroups of G then so also are U ∩ V and 〈 U, V 〉.