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If $n$ is prime there is only one isomorphism class of groups of order $n$, namely the cyclic group. I was wondering if there are other non-trivial examples of numbers $n$ where a non-trivial bound on the number of non-isomorphic groups of order $n$ can be proved. And how far the bounds can be extended for general $n$?

user2566092
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    For powers of primes, I think you have asymptotic formulas. For small powers, you have precise results. One group of order $p$, two groups of order $p^2$, five of order $p^3$, and of course the abelian case is easy with the structure theorem. – Pedro Apr 30 '14 at 21:35
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    For $n = p^a q^b$ you have Burnside's theorem which tells you that all such groups are solvable. So the problem reduces to bounding the number of extensions appearing in a composition series. Probably you don't get a great bound this way though. I think it's known that asymptotically almost all finite groups are solvable so maybe it's always possible to write down a bound like this. – Qiaochu Yuan Apr 30 '14 at 22:20
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    There are general upper bounds for all $n$: see Nick Gill's answer here: http://mathoverflow.net/questions/151491 – Derek Holt Apr 30 '14 at 23:00

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