On (n+1/2)-Engel groups

Let n be a positive integer. We say that a group G is an (n+1/2)-Engel group if it satisfies the law [x,yn,x]=1. The variety of (n+1/2)-Engel groups lies between the varieties of n-Engel groups and (n+1)-Engel groups. In this paper, we study these groups, and in particular, we prove that all (4+1/2)-Engel {2,3}-groups are locally nilpotent. We also show that if G is a (4+1/2)-Engel p-group, where p≥5is a prime, then G^p is locally nilpotent

A note on the local nilpotence of 4-Engel groups

MSC 20F45 Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T

An Engel condition for orderable groups

Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular case where v=x the group G is nilpotent (rather than merely locally nilpotent)

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