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On the Right and Left 4-Engel Elements

In this article we study left and right 4-Engel elements of a group. In particular, we prove that ⟨a, a b ⟩ is nilpotent of class at most 4, whenever a is of finite order and b ±1 are right 4-Engel elements or a ±1 are left 4-Engel elements and b is an arbitrary element of G. Furthermore, we prove that for any prime p and any element a of finite p-power order in a group G such that a ±1 ∈ L 4(G), a 4, if p = 2, and a p , if p is an odd prime number, is in the Baer radical of G.