We give an infinite family of examples that generalize the construction given by Noce, Tracey and Traustason a locally finite 2-group G containing left 3-Engel element x where 〈x〉G, normal closure in G, is not nilpotent. The based on Lie algebras are interest their own right make use classical theorem Lucas, regarding when (mn) even.

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological groups). We determine that any in compact-open topology on self-homeomorphisms Cantor set must be uniformly discrete, a precise sense we introduce here. Uniformly discrete are particular countable, finite, residually finite and topology. The resulting form subclass am...