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OP-9380OpenMathematics · general

Erdős #1160 · group theory

Let g(n)g(n) denote the number of groups of order nn. If n≤2mn\leq 2^m then g(n)≤g(2m)g(n)\leq g(2^m).

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

Let g(n)g(n) denote the number of groups of order nn. If n≤2mn\leq 2^m then g(n)≤g(2m)g(n)\leq g(2^m).

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Background

This is listed as an open problem (Question 22.16) in [BNV07], which reports it as a 'quite natural conjecture, whose origin we have been unable to trace satisfactorily. We have heard it attributed at various times to various people, such as Paul Erdős and Graham Higman.'

Question 22.18 of [BNV07] suggests the even stronger conjecture

∑n<2mg(n)≤g(2m)\sum_{n<2^m}g(n) \leq g(2^m)

for all sufficiently large mm (perhaps even as soon as m≥7m\geq 7).

Pantelidakis [Pa03] proved that the original conjecture is true if nn is odd and m≥3619m\geq 3619.

Progress & source assessment

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 #1160 link

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