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I read that there is a conjecture that most groups are $2$-groups.

This conjecture comes from the fact that by Higman-Sims asymptotic formula,

$\#$ of $p$-group of order $p^k= p^{\frac{2}{27}k^3 + O(\text{lower order terms})}$ with the fact that 2 is the smallest prime, so power of 2 just appears a lot more frequently. ($2^{100}\approx 3^{63} \approx 5^{43}$). But this conjecture seems to be far from being proven.

First, we have to formally define what "most" means. $\dfrac{\#_{\text{2-Group}}(\leq n)}{\#_{\text{Group}}(\leq n)}\to 1$ as $n\to \infty$ would be the formal statement of what it means for most groups to be $2$-groups.

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So I was wondering if there are weaker results?

  1. Most $p$-Groups are $2$-Groups. ie $\dfrac{\#_{\text{2-Group}}(\leq n)}{\#_{p\text{-Group}}(\leq n)}\to 1$ (It feels like this one is the easiest question, since it could be answered using just Higman-Sims formula along with some knowledge of the density of prime powers)

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  1. Most Groups are $p$-Groups
  2. Most Groups are Nilpotent
  3. Most Groups are Solvable

Are any of these results proven? I believe all of these questions are true heuristically.

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  • $\begingroup$ I doubt any of these is known. Consider comparing the number of groups of order $2^n$ with the number of groups of order $2^{n-6} \cdot 60$ of the form $P_2 \times A_5$. The Higman--Sims formula is not able to distinguish these counts, since $(n-6)^3 = n^3 - O(n^2)$. $\endgroup$ Nov 14, 2023 at 9:19
  • $\begingroup$ @SeanEberhard But obviously there are less non-solvable groups of order $2^{n-6}\cdot 60$ than groups of order $2^n$. So this is not an obstacle to prove that most groups are solvable in the sense of the OP. $\endgroup$ Nov 15, 2023 at 16:40
  • $\begingroup$ @BrauerSuzuki Why obviously? Intuitively, sure. The point is we know embarrassingly little about the number of groups of order $2^n$ as a function of $n$. $\endgroup$ Nov 15, 2023 at 16:43
  • $\begingroup$ By the Krull-Schmidt theorem the following map is injective: $P_2\times A_5\mapsto P_2\times C_{64}$. $\endgroup$ Nov 15, 2023 at 17:07

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There is an excellent book by Blackburn-Neumann-Venkataraman called "Enumeration of finite groups". At the very end, it lists many open questions. Question 22.19 (attributed to Pyber and Mann) is precisely your question "Are most groups nilpotent?". The book also gives some evidence, but in general it still seems to be open.

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