# Group in which no non-identity element has arbitrarily large roots

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

## Definition

A **group in which no non-identity element has arbitrarily large roots** is a group with the property that if is a non-identity element of , there exists some natural number (dependent on ) such that the equation has no solution.

## Facts

- Finitary symmetric group implies no non-identity element has arbitrarily large roots: This is used to prove that locally finite not implies embeddable in finitary symmetric group, because there are locally finite groups that don't have this property and they cannot be embedded in the finitary symmetric group, which has the property.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite group | Group embeddable in a finitary symmetric group, Residually finite group|FULL LIST, MORE INFO | |||

group of finite exponent | |FULL LIST, MORE INFO | |||

residually finite group | |FULL LIST, MORE INFO | |||

group embeddable in a finitary symmetric group | |FULL LIST, MORE INFO | |||

free group | Residually finite group|FULL LIST, MORE INFO |