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

From Groupprops

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## 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.

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

## Relation with other properties

### Stronger properties

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

finite abelian group | Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO | |||

odd-order class two group | ||||

finite Lazard Lie group | Finite group admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO |

### Weaker properties

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

finite nilpotent group |