= Solution
{c}
Suppose there were only finitely many <prime numbers>, say $p_1,\ldots,p_k$. Work in the <infinite group> $(\mathbb Z,+)$ and set $H_i=p_i\mathbb Z$. The <left cosets> are the $p_i$ residue classes modulo $p_i$, so each $H_i$ is a <finite-index subgroup>, with $|\mathbb Z/H_i|=p_i$.
By part (b)(v), $\mathbb Z\setminus\bigcup_i p_i\mathbb Z$ is empty or infinite. But part (a)(ii) identifies it as the nonempty two-element set $\{-1,1\}$. This contradiction proves <Euclid's theorem>:
$$
\boxed{\text{There are infinitely many prime numbers}.}
$$
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