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

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 no non-identity element has arbitrarily large roots is a group $G$ with the property that if $g$ is a non-identity element of $G$, there exists some natural number $n$ (dependent on $g$) such that the equation $x^{n}=g$ 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				|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				|FULL LIST, MORE INFO