Just for curiosity: given any $n\in\mathbb{N}$ how many non isomorphic groups are there? Is that number finite?countably infinite?uncountable? I know that depending on the $n$ it is finite for some cases though. Thank you for discussion.
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2If you mean groups of order $n$, then finite, there are only finitely many $n\times n$ multiplication tables. – André Nicolas Sep 12 '13 at 07:04
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1You have just run yourself into deep waters of group theory :) Assuming that you now believe there exist finitely many groups of a given finite order, take a look at here for more. – Prism Sep 12 '13 at 07:04
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1Also, consider taking a look at this useful page. – Prism Sep 12 '13 at 07:13
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Well, there is always finite since the number of maps $G\times G\to G$ is finite--so there are only finitely many groups of order $n$, let alone non-isomorphic groups of order $n$.
Alex Youcis
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3Strictly speaking there are infinitely many groups of order $1$. There's the trivial group on the set ${x}$ for any mathematical object $x$. In fact the collection of trivial groups does not even form a set! – Derek Holt Sep 12 '13 at 07:36
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The number of distinct abstract groups of given finite order $n$ is discussed in a very interesting paper by John H. Conway, Heiko Dietrich and E.A. O’Brien. Check the table at the end of the article (the number of groups for each order < 2048)! Observe that, as Tobias pointed out, the number of groups of prime power orders are substantially larger.
Nicky Hekster
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$2^83=1536$ is interesting. It has a factor of 10 fewer groups than $2^{10}=1024$ but (if we ignore $2^9=512$) a factor of 100 more than any other number on the list. – user1729 Sep 13 '13 at 09:09
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I have no idea! I would guess that it is because groups of the form $p^iq$ are "obviously" the next most plentiful after those of the form $p^i$, and $2^83$ is the number of the form $p^iq$ with the largest such $i$. This is the obvious generalisation of the conjecture that almost all groups are $p$-groups. But that is just a guess. – user1729 Sep 13 '13 at 13:25
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For most orders, it is not known how many groups there are of that order. It is expected (though nothing concrete has been proven that I know), that there tends to be many more groups of prime power orders than of other orders of the same approximate magnitude. For example, there are more than $50$ billion groups of order $1024$ and only about $100$ million groups of order at most $1023$.
http://www.math.ku.dk/~olsson/manus/three-group-numbers.pdf is a classification of those orders for which there are precisely $1$, $2$ or $3$ groups up to isomorphism. For larger numbers, the sort of arguments given there tend to become impractical.
Tobias Kildetoft
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