Problem 1163

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claims/: The 1 claim page of Problem 1163, one per claimant's result; the problem's standing derives from them.

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Statement. Describe (by statistical means) the arithmetic structure of the orders of subgroups of $S_n$.

Status. Open. The site's proof-claims tab carries one partial proof claim, a count of the subgroups of the symmetric group of a prescribed power-of-two order; the site shows no verdict and its label is unchanged (proof-claims thread accessed 2026-10-06). The claim is recorded on its claim page.

Source. erdosproblems.com/1163, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1163, https://www.erdosproblems.com/1163.

References.

• [Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Formalization. None recorded.

Current assessment

The question (site formulation of 2026-09-04). The statement above; OPEN. The site's commentary says that the problem is reproduced word for word from [Va99], where it is attributed to Erdős and Turán, and that the curator is not certain what it asks for. The question is a request for a description, so what would count as a complete answer is not fixed by the statement; this page records no interpretation of its own and no assessment of the mathematics, and no literature search beyond the site and this corpus's library was made.

Claims. One pending partial claim, a count of the subgroups of prescribed power-of-two order (2026-09-04, developed with OpenAI GPT-6, as the site's entry names the system), asserts that for $\lfloor n/4\rfloor\le j\le\lfloor n/2\rfloor$ the number of subgroups of $S_n$ of order $2^j$ has base-two logarithm $n^2/16+O(n\log n)$, uniformly in $j$. It counts subgroups of one range of orders and, as its author states, does not describe the distribution of the orders of a random subgroup; it is unreviewed and stays claimed, and the frontmatter standing stays open.

Related results. Roney-Dougal and Tracey's preprint (arXiv:2503.05416, unrefereed), carded as Roney-Dougal and Tracey 2025, proves statistical theorems about a uniformly random subgroup of $S_n$, the first of the readings listed below. For every $\nu<\tfrac12-\tfrac{\sqrt3}{4}$, almost every subgroup has a Sylow $2$-subgroup of order at least $2^{\nu n}$, so the $2$-adic valuation of its order is at least $\nu n$ with probability tending to one (Theorem 6), and almost every nilpotent subgroup is a $2$-group (Theorem 5). Its count of $p$-subgroups, $p^{n^2/(4p^2)}2^{O(n\log n)}$ (Theorem 2), gives for $p=2$ the upper half of the pending claim's estimate. The paper is not a claim on this problem: the site does not credit it here, and this page adopts no reading under which these theorems would settle anything.

Search scope, 2026-10-07. The site's problem page, commentary, discussion thread and proof-claims page, and the claimant's web page. The discussion thread holds two posts (1 March and 29 April 2026) proposing readings of the question, which this page lists as candidates and not as its own interpretation: the order of a subgroup chosen uniformly at random from all subgroups of $S_n$, with its logarithm, its $p$-adic valuations and its number of prime factors as the statistics; the set of orders of subgroups as a subset of the divisors of $n!$; and the order of the subgroup generated by a few random permutations. arXiv, Crossref, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) The intended question has not been pinned down, and beyond Roney-Dougal and Tracey's paper the literature on the orders of subgroups of $S_n$, including the work of Kovács and Praeger that the claim cites, was not searched. (2) The pending claim has no referee or reviewer and no manuscript record, and no check of its argument is recorded.