1
$\begingroup$

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 $A$?

  • 1
    Hint: How many abelian groups of order $p^4$ exist? ($p$ is a prime number) Finally use the decomposition in primary components.2011-08-15
  • 0
    There is only two abelian groups of order $p^{4}$, namely, $\mathbb Z_{p}\times \mathbb Z_{p}\times \mathbb Z_{p}\times \mathbb Z_{p}$, and $\mathbb Z_{p^{4}}$. So, if $|G|=2^{4}5^{4}$, then $G$ is isomorphic to $\(\mathbb Z_{2}\)^{4}\times\(\mathbb Z_{5}\)^{4}$, $\mathbb Z_{2^{4}}\times\(\mathbb Z_{5}\)^{4}$, $\(\mathbb Z_{2}\)^{4}\times\mathbb Z_{5^{4}}$, or $\mathbb Z_{2^{4}}\times\mathbb Z_{5^{4}}$.2011-08-15
  • 1
    Oh I forgot the cases $\mathbb Z_{p^{2}}\times \mathbb Z_{p^{2}}$, $\mathbb Z_{p^{2}}\times \mathbb Z_{p}\times \mathbb Z_{p}$, and $\mathbb Z_{p^{3}}\times \mathbb Z_{p}$, so we can proceed as above!2011-08-15
  • 1
    @William: What happened to $\mathbb{Z}_{p^2}\times\mathbb{Z}_{p^2}$, and to $\mathbb{Z}_{p^2}\times\mathbb{Z}_p\times \mathbb{Z}_p$?2011-08-15
  • 1
    Largest size of $A$? Does $A$ have a smaller size? Or more than one size for that matter?2011-08-15

1 Answers 1