Finite group that is 1-isomorphic to an abelian group

Definition
A finite group that is 1-isomorphic to an abelian group is a finite group that is 1-isomorphic to an abelian group (in particular, a defining ingredient::finite abelian group).

Two groups are 1-isomorphic if there is a 1-isomorphism between then: a bijection between them that restricts to an isomorphism of groups on each cyclic subgroup of either side.

Equivalently, a finite group that is 1-isomorphic to an abelian group is a finite nilpotent group for which each Sylow subgroup is a group of prime power order 1-isomorphic to an abelian group: it is 1-isomorphic to an abelian group of prime power order.

See also group that is 1-isomorphic to an abelian group.