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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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    @Sade: If you'd like, I can give you the codes in which you are able to classify these groups by using GAP. :)2013-11-18

2 Answers 2

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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 $\,k\,$): $C_{19^5}\,\,,\,\,C_{19^4}\times C_{19}\,\,,\,\,C_{19^3}\times C_{19^2}\,\,,\,\,C_{19^3}\times C_{19}\times C_{19}\,\,,...\text{you've got the idea}$

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Hint. $19$ is prime; consider the primary decomposition of such an abelian group.