---

Let $f(n)$ be the number of concrete subgroups of $S_n$, and sample uniformly from those subgroups, not from isomorphism classes. These author-recorded specializations by Plasma AI use Theorems 1, 2, 4, 5 and 6 of the held Roney-Dougal–Tracey preprint. Their statements and sampling convention were inspected; their proofs are not independently reviewed here. The conclusions below retain that source dependence.

A logarithmic asymptotic is not a relative asymptotic

With logarithms to base two, Theorem 1 states, for fixed positive $\alpha$ and an absolute $\beta$,

$$2^{n^2/16+\alpha n\log_2 n}\le f(n) \le2^{n^2/16+\beta n^{3/2}}.$$

Hence $\log_2 f(n)/n^2\to1/16$, but $f(n)/2^{n^2/16}\ge2^{\alpha n\log_2 n}\to\infty$. The leading logarithm cannot be exponentiated into relative error tending to zero. The separate relative-asymptotic claim recorded on Problem 1162 needs its own critical-model classification; it is not a consequence of these two inequalities. No examination of that classification is supplied here.

Fixed odd prime groups are rare

Theorem 2 and the preceding lower bound imply, for each fixed odd prime $p$,

$$\Pr(H\text{ is a }p\text{-group}) \le2^{-\delta_p n^2+O_p(n\log n)},\qquad \delta_p=\frac1{16}-\frac{\log_2p}{4p^2}>0.$$

The positivity follows from $\log_2p<p^2/4$ for $p\ge3$: it holds at three and $x^2/4-\log_2x$ is increasing for real $x\ge3$. This bounds being a $p$-group, not the probability that $p$ divides $|H|$. Its constants depend on $p$, so it asserts no union estimate over growing primes.

Conditioning cannot be dropped

Theorem 5 says that among nilpotent subgroups, the probability of being a 2-group tends to one. Theorem 4 says that on $n\equiv3\pmod4$, unconditioned nilpotence has probability bounded away from one. Since every 2-group is nilpotent, the same subsequence rules out the inference that almost every subgroup is a 2-group.

Theorem 6 does supply a genuine order statistic: for every $0\le\nu<1/2-\sqrt3/4$, the probability that $v_2(|H|)\ge\nu n$ tends to one. Under the unrestricted wording “a statistical theorem” this is a response to the second question of Problem 1162, at the source's standing. It gives no limit law for the entire order, the narrower formulation chosen on that problem page. Problem 1163 leaves the meaning of a complete statistical description unspecified; the sampling measure and statistic here are explicit. Understanding the odd part of $|H|$ remains a separate question, with no unconditional 2-group reduction from these statements.