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2.82 (1966)

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Can the class of groups with the $n$-th Engel condition $[x, \underbrace{y, \dots, y}_n] = 1$ be defined by identical relations of the form $u = v$, where $u$ and $v$ are words without negative powers of variables?

Contributor: A. I. Shirshov

Progress

This can be done for $n = 1, 2, 3$ (A. I. Shirshov, Algebra i Logika, 2, no. 5 (1963), 5–18 (Russian)).

This has also been done for $n = 4$ (G. Traustason, J. Group Theory, 2, no. 1 (1999), 39–46). As remarked by O. Macedońska, this is also true for the class of locally graded $n$-Engel groups because by (Y. Kim, A. Rhemtulla, in Groups–Korea '94, de Gruyter, Berlin, 1995, 189–197) such a group is locally nilpotent and then by (R. G. Burns, Yu. Medvedev, J. Austral. Math. Soc., 64 (1998), 92–100) such a group is an extension of a nilpotent group of $n$-bounded class by a group of $n$-bounded exponent; then a classical result of Mal’cev implies that such groups satisfy a positive law.

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