{c}

Suppose $H$ were finite, say $|H|=r$. There are at most $n$ <left cosets>, each with $r$ elements by part (i), and they partition $G$. Hence $G$ would be a <finite set> with at most $nr$ elements. This contradicts the assumption that $G$ is an <infinite group>. Therefore $\boxed{H\text{ is infinite}}$.

Solution

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 6 / b / ii / Solution

 Solution (source code)