Words of Engel type are concise in residually finite groups

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall\u2019s question remains wide open in the class of residually finite groups. In the present article we show that various generalizations of the Engel word are concise in residually finite groups

Strong conciseness in profinite groups

A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than continum many values in G, the verbal subgroup w(G) is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words have the property that the corresponding verbal subgroup is finite in a profinite group G whenever the word takes at most countably many values in G. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if w is one of the group words x^2, x^3, x^6, [x^3,y] or [x,y,y], then w is strongly concise in the class of all profinite groups

A note on strong conciseness in virtually nilpotent profinite groups

A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than 2ℵ0 values in G, the verbal subgroup w(G) is finite. In this paper, we prove that every group word is strongly concise in the class of virtually nilpotent profinite groups

On finite-by-nilpotent profinite groups

none2noLet γn = [x1, …, xn] be the nth lower central word. Suppose that G is a profinite group where the conjugacy classes xγn(G) contains less than 2ℵ0 elements for any x ∈ G. We prove that then γn+1 (G) has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group G is finite-by-nilpotent if and only if there is a positive integer n such that xγn(G) contains less than 2ℵ0 elements, for any x ∈ G.mixedDetomi E.; Morigi M.Detomi, E.; Morigi, M

Strongly generating elements in finite and profinite groups

Given a finite group G and an element g∈G, we may compare the expected number e(G) of elements needed to generate G and the expected number e(G,g) of elements of G needed to generate G together with g. We address the following question: how large can the difference e(G)−e(G,g) be? We prove that in general this difference can be arbitrarily large. For example for every positive integer n there exists a finite 2-generated group G such that e(G)≥n but e(G,g)≤5 for some g∈G. However, if the derived subgroup of G is nilpotent, then e(G)−e(G,g)≤11 for every g∈G

Strong conciseness of coprime and anti-coprime commutators

A coprime commutator in a profinite group G is an element of the form [x,\ua0y], where x and y have coprime order and an anti-coprime commutator is a commutator [x,\ua0y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than 2\u21350. Moreover, a profinite group G has finite commutator subgroup G\u2032 if the cardinality of the set of anti-coprime commutators in G is less than 2\u21350

Bounding the order of a verbal subgroup in a residually finite group

Let w be a group-word. Given a group G, we denote by w(G) the verbal subgroup corresponding to the word w, that is, the subgroup generated by the set G(w) of all w-values in G. The word w is called concise in a class of groups X if w(G) is finite whenever G(w) is finite for a group G is an element of chi. It is a long-standing problem whether every word is concise in the class of residually finite groups. In this paper we examine several families of group-words and show that all words in those families are concise in residually finite groups

Words of Engel type are concise in residually finite groups. Part II

This work is a natural follow-up of the article [5]. Given a group-word w and a group G, the verbal subgroup w.G/ is the one generated by all w-values in G. The word w is called concise if w.G/ is finite whenever the set of w-values in G is finite. It is an open question whether every word is concise in residually finite groups. Let w D w.x1; : : : ; xk/ be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word OEwq; ny is concise in residually finite groups (Theorem 1.2) while the word OEw; ny is boundedly concise in residually finite groups (Theorem 1.1)