TITLE: The name of a certain type of groups
QUESTION [5 upvotes]: What is the name (if any) given to groups satisfying:
$$\forall x,y,z\in G [xyx^{-1}=(zxz^{-1})y(zxz^{-1})^{-1}]$$
I understand this question might not be suitable here, but I really can't search easily for the answer to the above question.

REPLY [7 votes]: This can be rewritten as:
$$(zx^{-1}z^{-1}x)y = y(zx^{-1}z^{-1}x)$$
Note that $zx^{-1}z^{-1}x$ can be any element of the generator set for the commutator subgroup, and it means that the generators of the commutator must commute will all elements of $G$. That means that all elements of the commutator subgroup commute with all elements of $G$.
So the commutator subgroup is a subgroup of the center of the group, or $[G,G]\subseteq Z(G)$.
Per Derek Holt's comment, this is called a nilpotent group of class $2$, or $2$-step nilpotent for short. (It's not clear if $2$-nilpotent includes $1$-nilpotent, which are commutative, but the commutative groups are also obviously included.)
Basically, we can easily show that $1\unlhd [G,G]\unlhd G$ is a central series if and only if your condition is true.
And one last link makes it explicit.