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A Theorem on Finite Abelian GroupsHere's a nice theorem on finite abelian groups I came up with while I was rereading my old abstract algebra textbook. I'm sure it's nothing new, but it was new and surprising to me, and I felt like writing it down somewhere.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 U_{17}. (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.

See AlsoEuclidean Algorithm, The Favorite Things In Other Bases Multiplication in Base 10 Repunits @ May (2001) June (2008) 