Local finiteness is closed under extensions
= Local finiteness is closed under extensions
For $1\to N\to G\to Q\to1$ with $N,Q$ locally finite, take a finitely generated subgroup $L\leq G$. 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