Local finiteness is quotient-closed

Statement
Any quotient group of a locally finite group is also locally finite. In other words, if $$\varphi:G \to H$$ is a surjective homomorphism, and $$G$$ is locally finite, so is $$H$$.

Locally finite group
A group is termed locally finite if every finitely generated subgroup of it is finite.

Related facts

 * Local finiteness is subgroup-closed
 * Local finiteness is extension-closed

Proof
Given: A locally finite group $$G$$, a surjective homomorphism $$\varphi:G \to H$$.

To prove: If $$A$$ is a finite subset of $$H$$, $$\langle A \rangle$$ is finite.

Proof:


 * 1) Construction of a finite set $$B \subseteq G$$ such that $$\varphi(B) = A$$: Since $$\varphi$$ is surjective, we can pick, for each $$a \in A$$, an element $$b \in G$$ such that $$\varphi(b) = a$$. Making such a choice for each $$a \in A$$, we get a finite subset $$B$$ of $$G$$ such that $$\varphi(B) = A$$.
 * 2) $$\varphi(\langle B \rangle) = \langle \varphi(B) \rangle = \langle A \rangle$$: This follows from the fact that $$\varphi$$ is a homomorphism.
 * 3) $$\langle B \rangle$$ is a finite group (Given data used: $$G$$ is locally finite): Since $$G$$ is locally finite, and $$B$$ is a finite subset, $$\langle B \rangle$$ is a finite group.
 * 4) $$\langle A \rangle$$ is finite: By steps (3) and (4), $$\langle A \rangle$$ is the image of a finite group, and hence, is finite.