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Let say cardinality of set S is $n=|S|$.
We know that there are $n^{n^2}$ all binary operations on that set.
To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations there are on that finite set.
But also this set and given operation must satisfy specific axioms: closure, associativity, identity and invertibility.

So how find out how many different groups can be created on that finite countable set?

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What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $n\cdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.) –  Jim Belk Dec 6 '13 at 6:51
    
I need to know most likely in general case! –  IremadzeArchil19910311 Dec 6 '13 at 6:52
    
This is a very difficult question, and an active topic of research. Start here. The numbers you see in this list are not exactly what you are asking, as you are not identifying isomorphic groups, so your problem is even tougher. –  Andres Caicedo Dec 6 '13 at 6:53
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There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth. –  Jim Belk Dec 6 '13 at 7:00
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But usually, different groups mean non-isomorphic groups. –  Derek Holt Dec 6 '13 at 8:49

1 Answer 1

Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.): enter image description here

Instead of repeating what he says let me remark that that $\Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)

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