Left 3-engel elements of odd order in groups

Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w ³ = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x ³ _n +1 ^w ₃ ) ³ = 1.