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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 re-reading 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 Un 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 Un contain all possible finite abelian groups … or, more formally,
Every finite abelian group is isomorphic to a subgroup of Un 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.
Euclidean Algorithm, The
In Other Bases
Multiplication in Base 10
@ May (2001)