On groups with a locally nilpotent triple factorization (original) (raw)
Related papers
Groups with a nilpotent triple factorisation
Bulletin of the Australian Mathematical Society, 1988
In the investigation of factorised groups one often encounters groups G = AB = AK -BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup if of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that if is a minimax group or G has finite abelian section rank. The results become false if K has only finite Priifer rank. Some applications of the main theorems are pointed out.
On finite products of nilpotent groups
Archiv der Mathematik, 1994
i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubility of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addition, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below.
On Normal Subgroups of Products of Nilpotent Groups
Journal of the Australian Mathematical Society, 1988
Let G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.
On triple factorisations of finite groups
2009
This paper introduces and develops a general framework for studying triple factorisations of the form G = ABA of finite groups G, with A and B subgroups of G. We call such a factorisation nondegenerate if G = AB. Consideration of the action of G by right multiplication on the right cosets of B leads to a nontrivial upper bound for |G| by applying results about subsets of restricted movement. For A < C < G and B < D < G the factorisation G = CDC may be degenerate even if G = ABA is nondegenerate. Similarly forming quotients may lead to degenerate triple factorisations. A rationale is given for reducing the study of nondegenerate triple factorisations to those in which G acts faithfully and primitively on the cosets of A. This involves study of a wreath product construction for triple factorisations.
On Torsion-by-Nilpotent Groups
Journal of Algebra, 2001
Let C C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C C are torsion-by-nilpotent, then all soluble groups of C C are torsion-by-nilpotent. From that, we deduce the following conse-Ž quence, similar to a well-known result of P. Hall 1958, Illinois J. Math. 2,. 787᎐801 : if H is a normal subgroup of a group G such that H and GrHЈ are Ž. Ž. locally finite-by-nilpotent, then G is locally finite-by-nilpotent. We give an Ž. example showing that this last statement is false when '' locally finite-by-nilpotent'' is replaced with ''torsion-by-nilpotent.''
On the Nilpotency of a Pair of Groups
Southeast Asian Bulletin of …, 2012
This paper is devoted to suggest that the extensive theory of nilpotency, upper and lower central series of groups could be extended in an interesting and useful way to a theory for pairs of groups. Also this yields some information on nilpotent groups.
Introduction to Nilpotent Groups
The Theory of Nilpotent Groups, 2017
is an ascending series (or an ascending chain of subgroups). (ii) If G i G j for 1 Ä i Ä j; then G 1 G 2 G 3 (2.2) is a descending series (or a descending chain of subgroups). An ascending series may not reach G: If it does, then we say that the series terminates in G. Similarly, a descending series which reaches the identity is said to terminate in the identity. If there exists an integer m > 1 such that G m 1 ¤ G m and G m D G mC1 D G mC2 D in either (2.1) or (2.2), then the series is said to stabilize in G m. 2.1.2 Definition of a Nilpotent Group Definition 2.3 A group G is called nilpotent if it has a normal series
Glasgow Mathematical Journal
Abstarct Let γ n = [x1,…,x n ] be the nth lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that ∣gXn∣lem|{g^{{X_n}}}| \le m∣gXn∣lem for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.