Isomorphism Types

Question

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.

Answer 1

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.