Engel group

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Definition

A group $G$ is termed an Engel group or nil group or nilgroup, if, given any two elements $x,y\in G$, there exists a $n$ such that the iterated commutator:

$[[\dots [x,y],y],y],\dots ],y]=e$

where $e$ denotes the identity element, $[x,y]=xyx^{-1}y^{-1}$ denotes the commutator of $x$ and $y$, and $y$ occurs $n$ times.

If there exists a $n$ that works for all pairs of elements of $G$, then we say that $G$ is a $n$-Engel group. A $n$-Engel group, for some $n$, is termed a bounded Engel group. Note that sometimes the term Engel group is used for bounded Engel group.

Note if we instead define the commutator as $[x,y]=x^{-1}y^{-1}xy$ we get an equivalent definition.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally nilpotent group	every finitely generated subgroup is nilpotent			|FULL LIST, MORE INFO
nilpotent group				|FULL LIST, MORE INFO
bounded Engel group				|FULL LIST, MORE INFO
2-locally nilpotent group	subgroup generated by two elements is always nilpotent			|FULL LIST, MORE INFO

Examples and counterexamples

Finite groups

• All the nilpotent groups, which are equivalent to locally nilpotent groups for finite groups, are Engel groups.

• The smallest non-Engel finite group is symmetric group:S3. To see that, consider $x=(1\,3)$, $y=(2\,3)$. Define $x_{0}=x$, $x_{i}=[x_{i-1},y]$ for $i\geq 1$ and you will see that none of $x_{0},x_{1},x_{2},\dots$ are the identity permutation. (The values of $x_{0},x_{1},x_{2},\dots$ depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)