# Number of distinct groups of order n upto isomorphism, for a fixed integer n.

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

• How many binary operations can you think of in a set of $n$ elements? Forget group axioms for a while. – Jyrki Lahtonen Aug 13 '13 at 6:02
• 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 $n$. – 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 $G\times G\to G$.