Erdős Problem #1162

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

group theory

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A problem of Erdős and Turán.

Let $f(n)$ count the number of subgroups of $S_n$. Pyber [Py93] proved that\[\log f(n) \asymp n^2.\]Roney-Dougal and Tracey [RoTr25] have proved that\[\log f(n)=\left(\frac{1}{16}+o(1)\right)n^2.\]

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This page was last edited 23 January 2026.

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