# Group embeddable in a finitary symmetric group

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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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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

### 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.
View more such properties