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Classify, up to isomorphism, all abelian groups of order 2,000, giving the standard form of each group in your list. (The standard form is also called the invariant factor decomposition.)

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Eer....where in your question do characters appear?? – DonAntonio Oct 31 '12 at 22:31
sorry, I changed the title. – neno Nov 14 '12 at 17:47
up vote 6 down vote accepted

$$2,000=2^4\cdot 5^3$$

Now, the number of partitions of $\,4\,$ is $\,5\,$, and the number of partitions of $\,3\,$ is $\,3\,$ , so the number of different abelian groups of order $\,2,000\,$ up to isomorphism is $\,5\cdot 3=15\,$ .

Some of them are:

$$C_{2,000}\;,\;C_{16}\times C_{25}\times C_5\;,\;C_8\times C_2\times C_5\times C_5\ \times C_5\;,\;etc.$$

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Can I use Zn instead of Cn? – neno Nov 14 '12 at 17:43
I think so, but this can be a nuisance as the group operation changes to addition. But if you're more used to this no problem. – DonAntonio Nov 14 '12 at 18:42

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