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

Question

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

Thanks in advance.

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 06:02
@Jyrki Lahtonen: It is finite. I dont know exactly how many? — D. N., Aug 13 '13 at 06:08
Related: http://math.stackexchange.com/questions/422120/known-bounds-for-the-number-of-groups-of-a-given-order, http://math.stackexchange.com/questions/258753/a-set-of-non-isomorphic-finite-groups-is-a-finite-set — Jonas Meyer, Aug 13 '13 at 06:29
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 06:33

score 8 · Answer 1 · answered Aug 13 '13 at 06:02

8

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

answered Aug 13 '13 at 06:02

Alex Youcis

How can we know how many groups exist of a given order upto isomorphism? – low iq Jun 02 '16 at 09:18

score 7 · Accepted Answer · answered Aug 13 '13 at 06:03

7

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

answered Aug 13 '13 at 06:03

2 Answers2