Saturated formations and products of connected subgroups

Abstract

For a non-empty class of groups C, two subgroups A and B of a group G are said to be C-connected if a, b ∈ C for all a ∈ A and b ∈ B. Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π -subgroup by a ρ-group.It is shown that in a finite group G = A B, with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ (B) centralizes A O π (G)/O π (G), O ρ (A) centralizes B O π (G)/O π (G)and G ∈ S π ∪ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F -connected F -subgroups.

Products of Finite Connected Subgroups

“…In the present paper, as an application of Theorem 1, we show that the main results in [10][11][12], proved for soluble groups, remain valid for arbitrary finite groups. In particular, we characterize connected products for some relevant classes of groups (see Theorem 4).…”

“…[7][8][9]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [10,11]; in [12] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes.…”

A survey on Sylow normalizers and classes of groups

“…Finally, a study in [12] deals with the influence of 2-generator subgroups on the structure of groups and factorized groups. Here nilpotent-like Fitting formations of soluble groups are characterized like certain saturated formations F of soluble groups which are closed under taking products of F-connected F-subgroups.…”

“…For instance, given any sets π and ρ of primes, the class E π E ρ of all groups that are extensions of π-groups by ρ-groups is a nilpotent-like Fitting formation as well as the class of all groups with a Sylow tower for any given total ordering of the primes. Many more examples are described in [12].…”

2-Engel relations between subgroups

“…Therefore this paper can also be seen as a contribution to the study of C-connected subgroups for various classes of groups C (cf. [2], [6], [8], [9], [10], [14]). …”