Invariable generation of prosoluble groups

A group G is invariably generated by a subset S of G if G = \u3008sg(s) | s 08 S\u3009 for each choice of g(s) 08 G, s 08 S. Answering two questions posed by Kantor, Lubotzky and Shalev in [8], we prove that the free prosoluble group of rank d 65 2 cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank d and derived length l is invariably generated by precisely l(d 12 1) + 1 elements. \ua9 2016, Hebrew University of Jerusalem

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

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

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