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I'm a little panicking right now. I have finals soon, and I don't know how to go about solving this: Classify the isomorphism types of abelian groups of order 44. Solutions or even hints would be much appreciated.

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    Refer to the **fundamental theorem of finite abelian groups**. In particular, finite abelian groups decompose as direct sums of $p$-groups, and abelian $p$-groups are direct sums of cyclic groups of various $p$-power orders.2012-12-11
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    Questions very similar to this have been repeatedly asked, and answered. E.g. http://math.stackexchange.com/questions/111211/structure-theorem-for-finitely-generated-abelian-groups2012-12-11
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    I don't see how minus one is appropriate. Plus one.2012-12-11
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    Me too: I can't understand that minus one, so +12012-12-11

1 Answers 1

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From the Fundamental Theorem of f.g. abelian groups, one has that if we have the prime decomposition of $\,n\in\Bbb N\,$:

$$n=\prod_{k=1}^rp_k^{a_k}\,\,,\,\,p_k\,\,\text{primes}\,\,,\,\,a_k\in\Bbb N$$

Then the number of different abelian groups of order $\,n\, $ up to isomorphism is

$$\prod_{k=1}^r\mathcal P(a_k)\,\,\,,\,\,\,\mathcal P(a_k):=\, \text{number of different partitions of}\,\,a_k$$

Remember that a partition of a natural number is expressing it as a sum of natural numbers (I don't include zero as natural number), so for example $\,\mathcal P(2)=2\,\,,\,\,\mathcal P(4)=5\,\,\,,\,\,\mathcal P(6)=11$ , etc.

In your case we get $\,2\,$ different abelian groups of order $\,44\,$ up to isomorphism.

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    I think that $\rho(4)=5$. The partitions are: 1111,112,13,22,4.2013-02-18
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    Of course, thanks. That was a typo.2013-02-18