# Finite group that is 1-isomorphic to an abelian group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## 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 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.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions