Prove that contains a subgroup isomorphic to for all .
Let . We saw in a previous theorem that where .
Let , and let . Define by if is even and if is odd.
- is a homomorphism: If and are even, then . If without loss of generality is even and is odd, . If and are odd, then .
- is injective: Suppose . Then since is even and , we have . So the kernel of is trivial and is injective.
So embeds as a subgroup of . Moreover, recall that where . So this copy of in is generated by .