# Commutator map-equivalence of groups

## Definition

Suppose $G$ and $H$ are groups. A commutator map-equivalence of $G$ and $H$ is a bijection $f:G \to H$ such that for any $g_1,g_2 \in G$, we have:

$\! f([g_1,g_2]) = [f(g_1),f(g_2)]$

Note that it does not matter essentially for this definition whether we use the left or right convention for commutators, as long as we are consistent on both sides.

If there exists a commutator map-equivalence between two groups $G$ and $H$, we say that $G$ and $H$ are commutator map-equivalent groups.

## Facts

A commutator map-equivalence sends the identity element to the identity element. It induces a correspondence between the members of the upper central series of the two groups. In particular, it induces a bijection between the centers of the two groups, and they both have the same nilpotency class, so if one of them is a nilpotent group, so is the other one.