Finite group that is 1-isomorphic to an abelian group

From Groupprops
Jump to: navigation, search
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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