Commutator map-equivalent groups

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

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.

What determines a group up to commutator map-equivalence

 * Nilpotency class and order determine group up to commutator map-equivalence for up to prime-fourth order

Stronger relations

 * Isomorphic groups

Related relations

 * Isoclinic groups