# Commutator map-equivalent groups

From Groupprops

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Contents

## Definition

Two groups are termed **commutator map-equivalent groups** if there exists a bijection between them that is a commutator map-equivalence of groups.

## Facts

### Statistics determined up to commutator map-equivalence

The following are determined up to commutator map-equivalence:

- The order of the group.
- The orders of the members of the upper central series.
- Whether the group is a nilpotent group, and its nilpotency class if it is.
- Whether the group is a solvable group, and its derived length if it is.
- The number of elements of the group that can be expressed as commutators.
- The number of elements of the group that can be expressed as iterated commutators of any specified form.
- The multiset of orders of centralizers of elements of the group.
- The conjugacy class size statistics of the group.
- The commuting fraction of the group.