# Asymptotic density of certain class of finite groups (Solvable, Nilpotent, $p$-Group, etc).

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.

• 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 $(n-6)^3 = n^3 - O(n^2)$. Nov 14, 2023 at 9:19
• @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. Nov 15, 2023 at 16:40
• @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 $n$. Nov 15, 2023 at 16:43
• By the Krull-Schmidt theorem the following map is injective: $P_2\times A_5\mapsto P_2\times C_{64}$. Nov 15, 2023 at 17:07