Group in which elements of coprime finite orders commute
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 group in which elements of coprime finite orders commute is a group in which any two elements, both of which have finite orders such that the orders are relatively prime to others, must commute.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| nilpotent group | |FULL LIST, MORE INFO | |||
| locally nilpotent group | every finitely generated subgroup is nilpotent | |FULL LIST, MORE INFO | ||
| group in which order of commutator divides order of element | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group in which every finite subgroup is nilpotent | |FULL LIST, MORE INFO |