# Group embeddable in a finitary symmetric group

From Groupprops

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

## Contents

## 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.

View more such properties