Proportion of nonabelian $2$-groups of a certain order whose exponent is $4$

Let

$$\displaystyle A(n)=\frac{\text{number of nonabelian 2-groups of order $n$ whose exponent is }4}{\text{total number of nonabelian 2-groups of order $n$}}.$$

Using GAP, I could observe the following:

$$A(16)=\frac{5}{9}=0.5556, A(32)=\frac{21}{44}=0.4773, A(64)=\frac{93}{256}=0.3633, A(128)=\frac{820}{2313}=0.3545, A(256)=\frac{30446}{56070}=0.5430 \text{ and } A(512)=\frac{8791058}{10494183}=0.8377.$$

Can one prove that if $n>4$, then $A(n)>\frac{1}{3}$?

Solution 1:

It is not hard to prove a weaker result based using the seminal work of Sims (1961) and Higman (1964). Proving your result, I fear, will require a deeper understanding of $p$-groups than currently exists. Let $f(n,p)$ be the number of groups of order $p^n$ and let $f_2(n,p)$ be the number of such groups with $\Phi(G)=G'G^p$ central and elementary abelian. It follows from Sims that $f(n,p)\leqslant p^{cn^3+dn^{5/2}}$ (see [1]) and from Higman that $f_2(n,p)\geqslant p^{cn^3+en^2}$ where $c=\frac{2}{27}$ and $d$, $e$ are constants. Note that $d>0$. Higman had $e=-\frac{2}{9}$.

How does this relate to the ratio $A(n)$ in the above question? The groups of exponent dividing $p^2$ (and order $p^n$) are precisely counted by $f_2(n,p)$. Since a group of exponent 2 is elementary, we have $$A(2^n)=\frac{f_2(n,2)-1}{f(n,2)}\geqslant\frac{2^{cn^3+en^2}-1}{2^{cn^3+dn^{5/2}}}\to 0\qquad{\rm as}\ d>0.$$ Taking logarithms of the numerator and denominator above (and ignoring the $-1$), this ratio approaches 1, that is $\lim_{n\to\infty}\frac{cn^3+en^2}{cn^3+dn^{5/2}}=1$. Even if there were a constant $d'$ such that $f(n,p)\leqslant p^{cn^3+d'n^2}$ as Sims conjectured on p. 153. We must have $e<d'$ as $f_2(n,p)<f(n,p)$ and then $$\frac{f_2(n,2)}{f(n,2)}\geqslant\frac{2^{cn^3+en^2}}{2^{cn^3+d'n^{5/2}}}=2^{(e-d')n^2}\to 0\qquad{\rm as}\ e-d'<0.$$

Perhaps an expert on these error terms can say more.

[1] Simon R. Blackburn, Peter M. Neumann and Geetha Venkataraman,Enumeration of finite groups. Cambridge Tracts in Mathematics, 173. Cambridge University Press, Cambridge, 2007