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Dear all,

I am a student on computer science. So please forgive me if I state some results in a weird way, and I hope I ask an interesting question to you.

The question is related to finite groups with normal Hall subgroups. I want to know for groups of size $n$, what is the fraction of groups with normal Hall subgroups compared to all groups, up to isomorphism.

For example, we know that for given s, the number of non-isomorphic groups of size $n$ is bounded by $n^{O((\log n)^2)}$. While I can prove that for certain class of groups with normal Hall subgroup, for a given n, the number of non-isomorphic groups of size n can be $n^{\Omega(\log n)}$. But I would like to know an upper bound.

Thank you very much.

Jimmy

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  • $\begingroup$ I wonder if there is a meaningful question here that is not going to to be distorted by the prevalence of $p$-groups. For example they tell me about the 49487365422 groups of order 1024. $\endgroup$
    – Charles Matthews
    Commented Jul 29, 2010 at 8:43
  • $\begingroup$ I don't quite understand your statement, but I think I am asking for an asymptotic estimate. Of course it may be the case that for this question the fraction may not converge. Or I am asking "how many groups with at least one normal Hall subgroup are there for a given order", in the spirit of the classical question "how many groups are there for a given order". $\endgroup$
    – Jimmy
    Commented Jul 29, 2010 at 9:22
  • $\begingroup$ I think the growth rate for the total number of groups of order (at most) $n$ is more or less the same as just for $2$-groups, so requiring a normal Hall subgroup isn't such a strong restriction. One could however define a function $f(n)$ to be the number of groups of order at most $n$ with no normal Hall subgroups. Perhaps $f(n)$ grows roughly as fast as the total number of groups, perhaps not. This may be an interesting question. $\endgroup$
    – Colin Reid
    Commented Jul 29, 2010 at 9:49
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    $\begingroup$ @Jimmy: No it doesn't follow, because it also means there are a huge number of isomorphism classes that are (say) a product of a 2-group with a group of order 3. And those have Hall subgroups. The current question seems to be formulated in a way that doesn't seem to be sufficiently discriminating of cases that would be interesting. $\endgroup$
    – Charles Matthews
    Commented Jul 29, 2010 at 15:24
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    $\begingroup$ I like the questions: what fraction of groups of order p^r q^s have a normal Sylow p-subgroup. I tentatively think it would be "most" if p>q (and r,s are large enough). For n=1536, 98.9% have a normal Sylow 3, 2.5% have a normal Sylow 2, and 0.02% have no normal Sylow. $\endgroup$
    – Jack Schmidt
    Commented Jul 29, 2010 at 17:20

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