For any odd prime [Formula: see text], we give an example of a locally finite [Formula: see text]-group [Formula: see text] containing a left 3-Engel element [Formula: see text] where [Formula: see text] is not nilpotent.

We give an infinite family of examples that generalise the construction given in [10] of a locally finite 2-group G containing a left 3-Engel element x where x G , the normal closure of x in G, is not nilpotent. The construction is based on a family of Lie algebras that are of interest in their own right and make use of a classical theorem of Lucas, regarding when m n is even.

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