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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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 $H$.

Definitions used

Locally finite group

Further information: 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.