Group embeddable in a finitary symmetric 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 group embeddable in a finitary symmetric group is a group that can be expressed as a subgroup of the finitary symmetric group on some set.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| locally finite group | every finitely generated subgroup is finite | embeddable in finitary symmetric group implies locally finite | locally finite not implies embeddable in finitary symmetric group | |FULL LIST, MORE INFO |
| group in which no non-identity element has arbitrarily large roots | |FULL LIST, MORE INFO |
Metaproperties
Subgroups
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties
Direct products
This group property is restricted direct product-closed, viz., a restricted direct product of groups, each having the property, also has the property.
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