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OP-9381OpenMathematics · general

Erdős #1162 · group theory

Give an asymptotic formula for the number of subgroups of SnS_n. Is there a statistical theorem on their order?

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Give an asymptotic formula for the number of subgroups of SnS_n. Is there a statistical theorem on their order?

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Background

A problem of Erdős and Turán.

Let f(n)f(n) count the number of subgroups of SnS_n. Pyber [Py93] proved that

log⁡f(n)≍n2.\log f(n) \asymp n^2.

Roney-Dougal and Tracey [RoTr25] have proved that

log⁡f(n)=(116+o(1))n2.\log f(n)=\left(\frac{1}{16}+o(1)\right)n^2.

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1162 link

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