Local finiteness is quotient-closed
This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., quotient-closed group property)
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Locally finite group
Further information: Locally finite group
A group is termed locally finite if every finitely generated subgroup of it is finite.
Given: A locally finite group , a surjective homomorphism .
To prove: If is a finite subset of , is finite.
- Construction of a finite set such that : Since is surjective, we can pick, for each , an element such that . Making such a choice for each , we get a finite subset of such that .
- : This follows from the fact that is a homomorphism.
- is a finite group (Given data used: is locally finite): Since is locally finite, and is a finite subset, is a finite group.
- is finite: By steps (3) and (4), is the image of a finite group, and hence, is finite.