Listing all the albelian groups of order 900 A past exam question says: Using the fundamental theorem (of finitely generated albelian groups), list all the abelian groups of order 900 where no two groups in your list should be isomorphic but any group of order 900 must be isomorphic to a group in your list.
The list I have so far is
$$\mathbb{Z}_{900}$$
$$\mathbb{Z}_{3} \oplus \mathbb{Z}_{300}$$
$$\mathbb{Z}_{5} \oplus \mathbb{Z}_{180}$$
$$\mathbb{Z}_{15} \oplus \mathbb{Z}_{60}$$
$$\mathbb{Z}_{30} \oplus \mathbb{Z}_{30}$$
I have a started with the method of writing down all the factors of 900 but making sure that the factors aren't relatively prime, this is because if they were they would be isomorphic to $\mathbb{Z}_{900}$. I'm pretty sure I haven't got all the groups yet but I was just checking; is my method right? Is there a systematic method that I'm missing to get all the groups? (Also I haven't actually used the fundamental theorem and can't actually see how to use it.)
Thanks!

The systematic approach is to write down the factorisation $ 900 = 2^23^25^2 $. Then, for each prime factor, find all partitions of its exponent. Combine all these partitions to get the following list (I write $C_n$ instead of $\mathbb{Z}_n$ because I'm lazy):
$\begin{array}c
C_4 \times C_9 \times C_{25} ( \cong C_{900} ) ,\\
C_2 \times C_2 \times C_9 \times C_{25} ,\\
C_4 \times C_3 \times C_3 \times C_{25} ,\\
C_4 \times C_9 \times C_{5} \times C_5 ,\\
C_2 \times C_2 \times C_3 \times C_3 \times C_{25} ,\\
C_2 \times C_2 \times C_9 \times C_{5} \times C_5 ,\\
C_4 \times C_3 \times C_3 \times C_{5} \times C_5 ,\\
C_2 \times C_2 \times C_3 \times C_3 \times C_{5} \times C_5.
\end{array}$
The structure theorem for finitely generated abelian groups, also known as fundamental theorem, confirms that these are indeed all abelian groups of order $900$ and that they are pairwise non-isomorphic.
