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An Example 
The Structure of U_{n}The main thing to observe about the groups U_{n} is that they can be decomposed into prime power factors. Here's how that works … it all follows from one little fact. Suppose we can write n as a product m_{1}m_{2}, where m_{1} and m_{2} are relatively prime. It then so happens that if we know all about congruence mod m_{1} and m_{2}, we also know all about congruence mod n. In other words, two numbers a and b are congruent mod n,
a ≡ b mod n, if and only if they're congruent mod m_{1} and m_{2},
a ≡ b mod m_{1} Applying the rule repeatedly, we can break any group U_{n} into a product of groups of the form U_{pk}, where p is a prime and k is a positive integer. Now we just need to know what the groups U_{pk} look like. First, let's see how many elements they have. The elements, remember, are the numbers relatively prime to p^{k}, mod p^{k}. Now, being relatively prime to a number of the form p^{k} is the same as not being divisible by p; so to get the elements of the group, we just remove every p^{th} number from the p^{k} distinct numbers mod p^{k}:
o(U_{pk}) = (1  1/p) × p^{k} = (p1) p^{k1}. As for the internal structure of U_{pk}, well, I'll just summarize the results I found in Introduction to Number Theory.
That's everything I wanted to say about the groups U_{n}. However, there is one related point I'd like to make, which is that the cyclic groups Z_{n} can be decomposed into prime power factors, too. In fact, the decomposition follows from the same congruence argument as above, because any group Z_{n} can be thought of as a group of integers mod n, only under addition rather than multiplication. (Of course, the elements are no longer required to be relatively prime to n.) For example, we knew from the discussion above that the group U_{133} was the same as the cyclic group Z_{12 × 132}, but now we can further decompose the cyclic group into prime powers, like so.
U_{133} = Z_{12 × 132} = Z_{22} × Z_{3} × Z_{132}

See AlsoComplementary Parts Example, An (A Theorem on Finite Abelian Groups) Exceptions Explained, The Multiplication in Base 10 Multiplication in Base 7 Other Bases Proof, The Repeat Length Theorem on Finite Abelian Groups, A Usual Random Thoughts, The @ May (2001) 