Left $3$-Engel elements of odd order in groups

Let G be a group and let x epsilon 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(x(1),..., x(n)) 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(n+1)(3)w(3))(3) = 1.