Number of groups of a given order?

[tgt]

Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?

 

[adriank]

There isn't a general formula, but the number of groups has been tabulated for a large number of values.

http://people.csse.uwa.edu.au/gordon/remote/cubcay/ has a list of the number of groups up to order 1000. An interesting error causes it to say the number of groups of order 512 is -1, but it is actually 10,494,213.

Mathematica 7 includes the function FiniteGroupCount, which will tell you the number of groups of a given order, up to 2047.

 

[adriank]

I should add: In general, it's quite hard to find the number of groups of a given order. Can you prove that there are exactly 5 groups of order 8? 5 groups of order 12? It's not trivial.

Of course, for certain cases it's easy: Let p be a prime. Then there is exactly one group of order p (the cyclic one) and exactly two groups of order p² (there are two abelian ones for sure, and it's a bit harder to show that every group of order p² is abelian (hint: use the class equation)). It's harder to show that there are exactly five groups of order p³, but it's true.

 

[tgt]

This seems such a fundamental question that more emphasis should be put on it. Is it a famous open problem as groups are so widely used.