# Abelian Groups of order 2000

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

$$2,000=2^4\cdot 5^3$$
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 $\,5\cdot 3=15\,$ .
$$C_{2,000}\;,\;C_{16}\times C_{25}\times C_5\;,\;C_8\times C_2\times C_5\times C_5\ \times C_5\;,\;etc.$$