Group Laws Implying Virtual Nilpotence

Communications in Algebra, 2008

It is known that for any finitely generated group G from the large class of "locally graded" groups, satisfaction of an Engel or positive law forces G to be virtually nilpotent. In [2] Sarah Black gives a sufficient condition for an arbitrary 2-variable law w(x, y) ≡ 1 to imply virtual nilpotence-though only for finitely generated residually finite groups. We show how the Dichotomy Theorem from [4] for arbitrary words w(x 1 ,. .. , x n), encompasses Black's condition, extending it to the nvariable case and a certain large class S (however still falling short of the class of locally graded groups). We infer in particular that her condition is also necessary. We also deduce a simplified version of an algorithm of Qianlu Li [8, 9] for deciding whether or not a given law w(x 1 ,. .. , x n) ≡ 1 satisfies the extended version of Black's criterion.

Publicationes Mathematicae Debrecen, 2012

We answer the question: which property distinguishes the virtually nilpotent groups among the locally graded groups? The common property of each finitely generated group to have a finitely generated commutator subgroup is not sufficient. However, the finitely generated commutator subgroup of F2(var G), a free group of rank 2 in the variety defined by G, is the necessary and sufficient condition.

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 Group Theory, 2018

In this note we give two characterizations of finite nilpotent groups. First, we show that a finite group G is not p-nilpotent if and only if it contains two elements of order q k {q^{k}} , for q a prime different than p, whose product has order p or possibly 4 if p = 2 {p=2} . We also show that the set of words on two variables where the total degree of each variable is ± 1 {\pm 1} can be used to characterize finite nilpotent groups. Using this characterization we show that if a finite group is not nilpotent, then there is a word map of specified form for which the corresponding probability distribution is not uniform.

2018

The word w=[x_i_1,x_i_2,...,x_i_k] is a simple commutator word if k≥ 2, i_1≠ i_2 and i_j∈{1,...,m}, for some m>1. For a finite group G, we prove that if i_1≠ i_j for every j≠ 1, then the verbal subgroup corresponding to w is nilpotent if and only if |ab|=|a||b| for any w-values a,b∈ G of coprime orders. We also extend the result to a residually finite group G, provided that the set of all w-values in G is finite.

Canadian mathematical bulletin, 1994

Journal of Algebra, 1997

We investigate the structure of groups satisfying a positi¨e law, that is, an identity of the form u '¨, where u and¨are positive words. The main question here is whether all such groups are nilpotent-by-finite exponent. We answer this question affirmatively for a large class C C of groups including soluble and residually finite groups, showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law. It follows, in particular, that if a variety of groups is locally nilpotent-by-finite, then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent. We deduce various other corollaries, for instance, that a torsionfree, residually finite, n-Engel group is nilpotent of class bounded in terms of n. We also consider incidentally a question of Bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group, showing that it has an affirmative answer for soluble groups.

The paper concerns the problem of characteriza-tion of verbal subgroups in finitely generated nilpotent groups. We introduce the notion of verbal poverty and show that every verbally poor finitely generated nilpotent group is a finite -group with the lower -central series for certain prime . We conclude with few examples of verbally poor groups.

2021

Let G be a finite group, let p be a prime and let w be a group-word. We say that G satisfies P (w, p) if the prime p divides the order of xy for every w-value x in G of p-order and for every non-trivial w-value y in G of order divisible by p. With k ≥ 2, we prove that the kth term of the lower central series of G is p-nilpotent if and only if G satisfies P (γk, p). In addition, if G is soluble, we show that the kth term of the derived series of G is p-nilpotent if and only if G satisfies P (δk, p).

Journal of the Australian Mathematical Society, 1998

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.