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 > A Theorem on Finite Abelian Groups
Probability
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 The Structure of Un   The Proof   An Example

## A Theorem on Finite Abelian Groups

Here'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.

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

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See Also

Euclidean Algorithm, The
Favorite Things
In Other Bases
Multiplication in Base 10
Repunits

@ May (2001)
June (2008)