Local finiteness is closed under extensions

For $1\to N\to G\to Q\to 1$ with $N,Q$ locally finite, take a finitely generated subgroup $L\le G$. Its image in $Q$ is finite, so $L\cap N$ has finite index in $L$ and is finitely generated by Schreier's lemma. Local finiteness of $N$ makes that intersection finite. Thus $L$ itself is finite, proving local finiteness of $G$.

// Target: algebra.bigb