Given any positive integer $n$, is the number of distinct groups of order $n$ upto isomorphism finite?

Thanks in advance.

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    $\begingroup$ How many binary operations can you think of in a set of $n$ elements? Forget group axioms for a while. $\endgroup$ – Jyrki Lahtonen Aug 13 '13 at 6:02
  • $\begingroup$ @Jyrki Lahtonen: It is finite. I dont know exactly how many? $\endgroup$ – D. N. Aug 13 '13 at 6:08
  • $\begingroup$ deibor, so you were able to answer the question yourself! Good! Oh, and the number is just $n^{n^2}$ as the operation is just a function from a set of size $n^2$ to a set of size $n$. $\endgroup$ – Jyrki Lahtonen Aug 13 '13 at 6:33

Yes. Any such group is a subgroup of $S_n$, the symmetric group on $n$ elements. There are only finitely many such subgroups.


Yes, there are finitely many maps $G\times G\to G$.

  • $\begingroup$ How can we know how many groups exist of a given order upto isomorphism? $\endgroup$ – low iq Jun 2 '16 at 9:18

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