A note on Engel groups and local nilpotence

For a given positive integer n and a given prime number p, let r = r(n, p) be the integer satisfying p r−1 < n ≤ p r . We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)by-(finite exponent) where the best upper bound for the exponent is either p r or p r−1 if p is odd. When p = 2 the best upper bound is p r−1 , p r or p r+1 . In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.

International Journal of Algebra and Computation, 2005

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

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.''

2011

A subset S of a group G is called an Engel set if, for all x, y ∈ S, there is a non-negative integer n = n(x, y) such that [x, n y] = 1. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two.

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 $|{g^{{X_n}}}| \le m$ 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.

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

Journal of Group Theory, 2021

Following J. S. Rose, a subgroup 𝐻 of a group 𝐺 is said to be contranormal in 𝐺 if G = H G G=H^{G} . It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We study nilpotent-by-Černikov groups with no proper contranormal subgroups. Furthermore, we study the structure of groups with a finite proper contranormal subgroup.

Canadian mathematical bulletin, 1994

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.

Journal of the London Mathematical Society, 1982