I recently gave an undergraduate course on group theory (which is not entirely my field of expertise, so the following questions might have a well-known answer of which I am simply unaware). As I was explaining the concept of solvability, I digressed a little and told the class about the odd-order theorem, also known as the Feit-Thompson theorem, which states that every finite group of odd order is solvable. I made the remark: Among finite groups, solvability is the rule rather than the exception, because solvability is at least as likely as oddity. One of my students asked: "So if I take an arbitrary finite group, how likely is it then that this group is of odd order?" To which I knew no reply.


So I would like to ask the following series of related questions:


(1.) If \begin{equation\*}x\_{n}=\frac{\#\text{Isomorphy classes of groups of even order $\leq n$}}{\#\text{Isomorphy classes of groups of order $\leq n$}}\end{equation\*} does the series $x\_{n}$ converge? If not, what are its cluster points?


(2.) If $m\in\mathbb{N}$ and \begin{equation\*}y\_{n}=\frac{\#\text{Isomorphy classes of groups of order $\leq n$, divisible by $m$}}{\#\text{Isomorphy classes of groups of order $\leq n$}}\end{equation\*} does the series $y\_{n}$ converge? If not, what are its cluster points?


(3.) If \begin{equation\*}z\_{n}=\frac{\#\text{Isomorphy classes of solvable groups of order $\leq n$}}{\#\text{Isomorphy classes of groups of order $\leq n$}}\end{equation\*} does the series $z\_{n}$ converge? If not, what are its cluster points?


My simple intuition is that in all three cases, the answer should be "yes, it converges", and it should converge to $\frac{1}{m}$ in case (2.), and to a value $\geq\frac{1}{2}$ in case 3.


I beg your forgiveness in advance if the answers are well-known, I am not an expert on group theory.



Does the sequence (Number of groups of even order $\le n$) / (Number of groups of order $\leq n$) converge? If not, what are its cluster points?
As mentioned in the comments, conjecturally almost all finite groups are $2$-step nilpotent $2$-groups, so conjecturally the answers to 1) and 3) are that the limits both exist and both equal $1$; that is, almost all finite groups have even order and almost all finite groups are solvable (even nilpotent). As numerical evidence for this, [almost all of the first $50$ billion groups have order $1024$](https://golem.ph.utexas.edu/category/2012/11/almost_all_of_the_first_50_bil.html). The conjectural answer to 2) is then that if $m$ is a power of $2$ then the limit is equal to $1$ and otherwise if $m$ has a nontrivial odd divisor then the limit is equal to $0$.


It's worth knowing as context here that a result due to Higman and Sims states that asymptotically the number of $p$-groups of order $p^n$ is $p^{ \frac{2}{27} n^3 + O \left( n^{8/3} \right)}$. The lower bound comes from counting $2$-step nilpotent $p$-groups; you can see an analogous argument for nilpotent Lie algebras [here](https://mathoverflow.net/questions/371146/how-many-lie-and-associative-algebras-over-a-finite-field-are-there/371150#371150). Thinking of this count as a function of the order $p^n$ it's not hard to check that it's maximized, if $p^n$ is bounded by some reasonably large $N$, by making $p$ as small as possible (equivalently, by making $n$ as large as possible), which is what singles out $p = 2$. It should be possible to write down a similar heuristic argument showing that the count of nilpotent groups (which are products of their Sylow subgroups) is dominated by groups of order $2^n$ also.


