A Theorem on Finite Abelian Groups

Let U_n be the group of integers relatively prime to n, under multiplication mod n. So, for example, the equation

5 × 7 = 35 ≡ 1 mod 17

shows that 7 is the inverse of 5 in U₁₇. (That second equals sign is as close to a congruence sign as I could get.)

The theorem, then, is that the groups U_n contain all possible finite abelian groups … or, more formally,

Every finite abelian group is isomorphic to a subgroup of U_n for some appropriate n.

The result is of the same form as Cayley's theorem, which concerns the permutation groups A(S) (where S is a set of elements).

Every group is isomorphic to a subgroup of A(S) for some appropriate S.

I've divided the rest of the argument into three parts: first, some handy facts about the structure of U_n; second, a proof of the theorem, and finally an example worked out in detail.