How can I find all natural numbers so that there are (up to isomorphism) exactly 10 abelian groups of order n?

How can I use the main theorem about finite abelian groups to determine those numbers?

  • $\begingroup$ A finite abelian group is isomorphic to the direct product of its Sylow subgroups. $\endgroup$
    – user441558
    Nov 23, 2017 at 9:48
  • 1
    $\begingroup$ Find a formula for the number of abelian groups of order $n$ in terms of the prime factorization of $n$. $\endgroup$ Nov 23, 2017 at 9:49
  • $\begingroup$ Start by finding out how many abelian group there are of order $p^a$ for prime $p$. $\endgroup$ Nov 23, 2017 at 10:13
  • $\begingroup$ @GerryMyerson isn't that the number of partitions? For n =1 it's 1, for n=2 it's 2, for n=3 it's 3, for n=4 it's 5... And so on? $\endgroup$ Nov 23, 2017 at 10:18
  • 1
    $\begingroup$ There are only so many ways to multiply together vaues of the partition function to get 10. $\endgroup$ Nov 24, 2017 at 5:58


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