What is this the name of this property of groups?

Question

Let $G$ be a group such that for every $g,h\in G$, $[g,h^{-1}gh]=1$; that is, every element commutes with all its conjugates.

My questions:

1. Is there any name for such a property?
2. What are some examples of groups (where $G$ is non-Abelian) that satisfy this property?

Edit: I think that the discrete Heisenberg group $H_3(\mathbb{Z})=\langle x,y\mid [x,[x,y]]=[y,[x,y]]=1\rangle$ might be such an example but I am not quite sure.

Answer 1

These are the $2$-Engel groups. Note that $$ \mathbf{1 =\ } [g, h^{-1} g h] = [g, g [g, h]] = [g, [g, h]] [g,g]^{[g, h]} = \mathbf{[g, [g, h]]}. $$ So these are the group where for all $g, h \in G$ we have $$ [[h,g]], g] = 1. $$

Levi proved in

F. W. Levi, Groups in which the commutator operations satisfy certain algebraic conditions, J. Indian Math. Soc. 6 (1942) 87-97

that in a $2$-Engel group we have $[[a, b], c]^3 = 1$ for all $a, b, c$. So a $2$-Engel group is nilpotent of class at most $2$, unless it has $3$-torsion. And in this case one can show that it has class at most $3$.

See also groupprops.

Examples

• Nilpotent groups of class $2$, of course.
• Slightly less trivially, the free group of exponent $3$ in $3$ generators is $2$-Engel, and has class $3$. Note that exponent $3$ implies $2$-Engel.