# Isomorphism Types

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. –  anon Dec 11 '12 at 11:28
Questions very similar to this have been repeatedly asked, and answered. E.g. math.stackexchange.com/questions/111211/… –  Andrea Mori Dec 11 '12 at 11:32
I don't see how minus one is appropriate. Plus one. –  Rudy the Reindeer Dec 11 '12 at 11:33
Me too: I can't understand that minus one, so +1 –  DonAntonio Dec 11 '12 at 20:48

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. –  Ludolila Feb 18 '13 at 10:15
Of course, thanks. That was a typo. –  DonAntonio Feb 18 '13 at 10:48