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.
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 | |FULL LIST, MORE INFO | |||
| odd-order class two group | ||||
| finite Lazard Lie group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite nilpotent group |