On groups covered by locally nilpotent subgroups

Let N be the class of pronilpotent groups, or the class of locally nilpotent profinite groups, or the class of strongly locally nilpotent profinite groups. It is proved that a profinite group G is finite-by-N if and only if G is covered by countably many N-subgroups. The commutator subgroup G\ue2\u80\ub2is finite-by-N if and only if the set of all commutators in G is covered by countably many N-subgroups. Here, a group is strongly locally nilpotent if it generates a locally nilpotent variety of groups. According to Zelmanov, a locally nilpotent group is strongly locally nilpotent if and only if it is n-Engel for some positive n

Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G. This is obtained as a consequence of a structure theorem for the subgroup w(G), which is valid if G is soluble, and without assuming that w takes finitely many values in G. More precisely, there is an abelian series of w(G), such that every section of the series can be generated by values of w all of whose powers are also values of w in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction

Commutators and pronilpotent subgroups in profinite groups

Let G be a profinite group in which all pronilpotent subgroups generated by commutators are periodic. We prove that G' is locally finite

Bounding the Exponent of a Verbal Subgroup

We deal with the following conjecture. If w is a group word and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only. We show that this is true in the case where w is either the nth Engel word or the word [x^n,y_1,y_2,...,y_k] (Theorem A). Further, we show that for any positive integer e there exists a number k=k(e) such that if w is a word and G is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B)

The solvability of groups with nilpotent minimal coverings

A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a proof that every group that has a nilpotent minimal covering is solvable, starting from the previously known result that a minimal counterexample is an almost simple finite group

On profinite groups with commutators covered by countably many cosets

Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union of countably many cosets giGi, where each Gi is in C, then the verbal subgroup w(G) is virtually-C. This strengthens several known results

Lower central words in finite p-groups

It is well known that the set of values of a lower central word in a group G need not be a subgroup. For a fixed lower central word γr and for p ≥ 5, Guralnick showed that if G is a finite p-group such that the verbal subgroup γr(G) is abelian and 2-generator, then γr(G) consists only of γr-values. In this paper we extend this result, showing that the assumption that γr(G) is abelian can be dropped. Moreover, we show that the result remains true even if p= 3. Finally, we prove that the analogous result for pro-p groups is true