Commutator map-equivalent groups

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

Definition

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

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.