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Define a set

$A:=\{G: G$ is abelian group; $\operatorname{order}(G)=10.000$, no two groups are isomorphic $\}$.

What is the largest size of $A$?

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    Largest size of $A$? Does $A$ have a smaller size? Or more than one size for that matter?2011-08-15

1 Answers 1

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This is an easy application of the Fundamental Theorem of Finitely Generated Abelian Groups. Since $|G|=2^4\times 5^4$, then $G\cong G_2\times G_5$, where $G_p$ is the $p$-part of $G$.

Thus, the problem reduces to asking how many different abelian groups of order $p^4$ there are. Since an abelian $p$-group must be a direct sum/product of cyclic groups of prime power order, the answer is that there are as many abelian groups of order $p^n$ as there are partitions of $n$.

So there are as many possibilities for $G_2$ as there are partitions of $4$, and also as many possibilities for $G_5$ as there are partitions of $4$, and hence the number will be the product of these two, the square of the number of partitions of $4$.

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    Sure, we will have 2+1+1- I already mentioned all of them above in my first and second comments-typo!, so its $25$ not $16$.2011-08-15