Left 3-Engel Elements in Locally Finite 2-Groups

Abstract

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.