# gr.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?

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 $$minmathbb{N}$$ 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 $$geqfrac{1}{2}$$ in case 3.

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