Maybe the question (and particularly title) is somewhat silly. Anyway, one can observe that variety of 2-Engel groups may be alternatively described as variety of groups which have skew associative commutator operation: $\mathfrak B ([x, y, y]) \equiv \mathfrak B ([[x, y], z][x, [y, z]])$.

  1. Can we define n-Engel groups through "higher associators" identities?

  2. Concretely, is multiplicative pentagonator identity of the form $$[x, y, z, w]^{\epsilon}[x, [y, z], w]^{\epsilon}[x, [y, z, w]]^{\epsilon}[x, [y, [z, w]]]^{\epsilon}[[x, y], [z, w]]^{\epsilon}$$ with some choice of $\epsilon 's = \pm 1$ defining for 3-Engel groups? (it obviously implies 3-Engelness for any choice of exponents).

If not, maybe those free pentagon groups will share some local properties with free 3-Engel; every 3-Engel group is locally nilpotent.

  1. Are pentagon groups nilpotent?

  2. Are their 4-generator subgroups nilpotent of fixed degree? For example, groups where operations [x, y] and [y, x] distribute over each other are 3-locally metabelian but may fail to be metabelian with minimal counterexample being something like 2-Sylow subgroup of $S_{14}$.

I think (but it's too lousy for actual statement) that possibility of associator presentations is related to gaps (or their absence) between conditions in this chain of implications:

(a) $\langle x\rangle^G$ are $n$-nilpotent $\Rightarrow$ (b) $\langle x\rangle^G$ are $n$-Engel $\Rightarrow$ (c) $G$ is $(n+1)$-Engel

(c) implies (a) for $n \le 2$ but up to my knowledge equivalence is unknown for higher $n$'s. (see edit)

  1. What is known today about normal closures of elements in n-Engel groups?

Edit: N. Gupta and F. Levin constructed in [1] infinitely generated counterexample where (c) $\not \Rightarrow$ (a) for $n = \text{odd prime} + 1$: take countably generated free 2-nilpotent exponent $p$ group $H$ and let $G$ be the semidirect product $\mathbb F_p[H] \rtimes H$ via regular representation. Then $G$ is $p+2$-Engel, but normal closure of $(1, x)$ is not nilpotent for $x$ a generator of $H$.

In same paper they provide intricate example of finitely generated 4-Engel group where normal closures of elements have nilpotence degree 4.

// I've also reformulated and listed subquestions for comprehensibility.

[1] N. Gupta and F. Levin, On soluble Engel groups and Lie algebras, Arch. Math. 34 (1980)


For Quesion 5:

What is known today about normal closures of elements in n-Engel groups?

see the following papers:

Traustason, Gunnar, Locally nilpotent 4-Engel groups are Fitting groups. J. Algebra 270, No. 1, 7-27 (2003).

Vaughan-Lee, Michael, On 4-Engel groups. LMS J. Comput. Math. 10, 341-353 (2007).

In the latter paper it is proved that a group is a 4-Engel group if and only if the normal closure of every element is a 3-Engel group.


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