2-Engel relations between subgroups

Abstract

In this paper we study groups G generated by two subgroups A and B such that ⟨a, b⟩ is nilpotent of class at most 2 for all a ∈ A and b ∈ B. A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.

Nilpotence relations in products of groups

“…In the literature, the results about N -connection mainly concern products of finite groups. An example where the authors did not assume G to be a product is in [27]. However, we are able to find examples that, in our opinion, discourage the continuation of the research in that direction.…”

“…As far as the N 2 -connection is concerned, the main results are to be found in [27]. This connection is in some sense the strongest we have seen until now and one of the few in literature that is treated without assuming the finiteness of the involved groups.…”

Products of Finite Connected Subgroups

“…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 study in [13] of connected subgroups, for the class of finite nilpotent groups of class at most 2, contributes generalizations of the classical results on 2-Engel groups.…”

Thompson-like characterization of solubility for products of finite groups

“…For the special case when G = AB = A = B this means of course that a, b ∈ L for all a, b ∈ G, and the study of products of L-connected subgroups provides a more general setting for local-global questions related to two-generated subgroups. We refer to [8,28,9] for previous studies for the class L = N of finite nilpotent groups, and to [18,19,20,21] for L being the class of finite metanilpotent groups and other relevant classes of groups. For the class L = S of finite soluble groups, A. Carocca in [12] proved the solubility of a product of S-connected soluble subgroups, which provides a first extension of the above-mentioned theorem of Thompson for products of groups (see Corollary 2).…”