# Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?

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Hint:Every finite Abelian group is a direct product of cyclic groups – Geoff Robinson Jul 21 '12 at 20:10
Can you think of any abelian groups of order $19^5$? – Zev Chonoles Jul 21 '12 at 20:10
I must ask you @Patience, how long have you worked on this question? And more generally, what is your backgroud? You are putting out an awful lot of questions on a broad range of topics; unless you've spent some time on them you won't profit from the solution. You'll know the basic framework of how to solve them, but if my experience has taught me anything, you'll quickly forget, and in the long run you might not retain very much of it at all. For this question in particular: do you know the structure theorem for finite(ly generated) abelian groups? – Olivier Bégassat Jul 21 '12 at 20:20
@Patience: No. Because two such products are isomorphic if and only if they are identical (up to order of the factors), and there are many ways of writing $19^5$ as a product of prime powers. In fact, there are seven nonisomorphic abelian groups of order $19^5$. – Arturo Magidin Jul 21 '12 at 20:36
No, they are non isomorphic – La Belle Noiseuse Jul 21 '12 at 20:41

From the Fundamental Theorem for Fin. Gen. Abelian groups, it follows that we must take the partitions of 5 (all this can be googled easily): \begin{align*}5=&5\\5=&4+1\\5=&3+2\\5=&3+1+1\\5=&2+2+1\\5=&2+1+1+1\\5=&1+1+1+1+1\end{align*} Since there are 7 such partitions, there are 7 non-isomorphic groups of order $\,19^5\,$, which are (notation: $\,C_k=\,$ the cyclic group of order $\,k\,$): $$C_{19^5}\,\,,\,\,C_{19^4}\times C_{19}\,\,,\,\,C_{19^3}\times C_{19^2}\,\,,\,\,C_{19^3}\times C_{19}\times C_{19}\,\,,...\text{you've got the idea}$$
Hint. $19$ is prime; consider the primary decomposition of such an abelian group.