Local finiteness is extension-closed

This article gives the statement, and possibly proof, of a group property (i.e., locally finite group) satisfying a group metaproperty (i.e., extension-closed group property)
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Statement

Statement with symbols

Suppose $G$ is a group with a normal subgroup $H$ such that both $H$ and $G/H$ are locally finite groups: any finitely generated subgroup of $H$ is finite, and any finitely generated subgroup of $G/H$ is finite. Then, $G$ is itself locally finite.

Related facts

• Periodicity is extension-closed
• Finite generation is extension-closed

Facts used

1. Schreier's lemma: In its factual form, this states that any subgroup of finite index in a finitely generated group is again finitely generated.
2. Finiteness is extension-closed
3. First isomorphism theorem

Proof

Given: A group $G$ with a normal subgroup $H$ such that both $H$ and $G/H$ are locally finite. Let $\varphi :G\to G/H$ be the quotient map.A finite subset $A$ of $G$ with $K=\langle A\rangle$.

To prove: $K$ is finite.

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$\varphi (K)$ is a finite subgroup of $G/H$		$G/H$ is locally finite		Since $A$ is finite, $\varphi (A)$ is finite. By local finiteness of $G/H$, $\langle \varphi (A)\rangle$ is a finite subgroup of $G/H$. We have $\langle \varphi (A)\rangle =\varphi (\langle A\rangle )=\varphi (K)$, so $\varphi (K)$ is finite.
2	$K\cap H$ is a normal subgroup of finite index in $K$	Fact (3)			We have $\varphi (K)\cong K/(K\cap \ker \varphi )=K/(K\cap H)$. Since the left side is finite, we conclude that $K\cap H$ has finite index in $K$.
3	$K\cap H$ is finitely generated	Fact (1)	$A$ is finite	Step (2)	$A$ is finite, so $K=\langle A\rangle$ is finitely generated, and by step (2), $K\cap H$ has finite index in $K$. Thus, by fact (1), $K\cap H$ is also a finitely generated group.
4	$K\cap H$ is finite		$H$ is locally finite	Step (3)	$K\cap H$ is a finitely generated subgroup of $H$. Local finiteness of $H$ forces $K\cap H$ to be finite.
5	$K$ is finite	Fact (2)		Steps (2), (4)	By step (4), $K\cap H$ is finite, and by step (2), $H/(H\cap N)$ is finite. Thus, by fact (2), $K$ is finite.

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