GPT 6 Astra Ultra attempt on Math-6216

Erdős Problem 1162 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

For the symmetric group $S_n$, seek an asymptotic formula for the number $|\operatorname{Sub}(S_n)|$ of its subgroups. Also seek a statistical description of the orders of those subgroups.

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

The question asks for an asymptotic formula for the subgroup count, not merely its leading logarithmic exponent. Its statistical clause does not specify a probability model, so none is imposed here.

ATTACK PLAN AND WORK

Put $k=\lfloor n/2\rfloor$. The disjoint transpositions $(1\ 2),(3\ 4),\ldots,(2k-1\ 2k)$ generate a subgroup $E\cong\mathbb F_2^k$ of $S_n$. Split $E=U\oplus V$ with $\dim U=\lfloor k/2\rfloor$ and $\dim V=\lceil k/2\rceil$. Every linear map $T:U\to V$ gives a distinct graph subgroup $\{(u,Tu):u\in U\}$. Consequently \[|\operatorname{Sub}(S_n)|\ge2^{\lfloor k^2/4\rfloor} =2^{n^2/16-O(n)}.\tag{1}\] All of these many subgroups even have the same order $2^{\lfloor k/2\rfloor}$.

There is also a crude universal upper bound requiring no permutation-group structure. Every subgroup of a finite group $G$ has a generating set of size at most $\lfloor\log_2|G|\rfloor$: whenever a chosen element enlarges the subgroup generated so far, its order at least doubles. Pad shorter generating lists with identities. Counting all lists then yields \[|\operatorname{Sub}(S_n)|\le(n!)^{\lfloor\log_2(n!)\rfloor} \le2^{n^2(\log_2 n)^2}.\tag{2}\] Neither $1$ nor $2$ is presented as a new or sharp literature bound. They are independent baseline proofs identifying a large explicit family and the very substantial precision still missing from this attempt.

VERIFICATION AND REMAINING GAP

Determine an asymptotic formula rather than bounds on different quadratic logarithmic scales. Counting one elementary abelian family also does not describe the order distribution of all subgroups of $S_n$.

SOURCES

https://www.erdosproblems.com/1162

https://www.erdosproblems.com/latex/1162

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$7\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.