Pairwise -connected Products of Certain Classes of Finite Groups

Abstract

Subgroups A and B of a finite group are said to be N-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and N-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.

Products of Finite Connected Subgroups

“…The structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [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.…”

Thompson-like characterization of solubility for products of finite groups

“…One can assume that m ≥ 4 since Sp 4 (2) = L 2 (9). Then r divides |A ∩ N | and t divides |B ∩ N | where r and t are as in Notation 3, or t = 7 if m = 4. r and t are independent.…”

“…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 [8,29,9] 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).…”

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).…”