Right 2-Engel Elements and Commuting Automorphisms of Groups

It is shown that there is a close connection between the right 2-Engel elements of a group and the set of the so-called commuting automorphisms of the group. As a consequence, the following general theorem is proved: If G is a group and if R2(G) denotes the subgroup of right 2-Engel elements, then the factor group R2(G) ∩ CG(G′)/Z2(G) is a group of exponent at most 2.