On the structure of some locally nilpotent groups without contranormal subgroups

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.

The complexity of ascendant sequences in locally nilpotent groups

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0, x1, …, xN ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly [Formula: see text]) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x0, x1, …, xN〉G, the subgroup generated by x0, x1, …, xN in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

RIGHT 4-ENGEL ELEMENTS OF A GROUP

We prove that the set of right 4-Engel elements of a group G is a subgroup for locally nilpotent groups G without elements of orders 2, 3 or 5; and in this case the normal closure ⟨x⟩G is nilpotent of class at most 7 for each right 4-Engel elements x of G.