Problem 1160

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claims/: The 1 claim page of Problem 1160, one per claimant's result; the problem's standing derives from them.

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Statement. Let $g(n)$ denote the number of groups of order $n$. If $n\leq 2^m$ then $g(n)\leq g(2^m)$.

Status. Open. The site's commentary credits Pantelidakis [Pa03] with the case of odd $n$ and $m\ge3619$, recorded as a pending partial claim on its claim page; it does not change the standing.

Source. erdosproblems.com/1160, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1160, https://www.erdosproblems.com/1160.

References.

• [BNV07] Blackburn, Simon R. and Neumann, Peter M. and Venkataraman, Geetha, Enumeration of finite groups. (2007), xii+281.
• [Pa03] I. Pantelidakis, On the Number of Non-isomorphic Groups of the Same Order. DPhil Thesis, University of Oxford (2003).

Formalization. None recorded.

Current assessment

The standing judges the site's formulation, and it is open. Blackburn, Neumann and Venkataraman [BNV07] list the statement as an open problem (Question 22.16), a natural conjecture whose origin they could not trace, attributed at various times to Erdős and to Graham Higman, among others. Their Question 22.18 proposes the stronger inequality $\sum_{n<2^m}g(n)\le g(2^m)$ for all sufficiently large $m$, perhaps from $m=7$ on.

The site's commentary credits Pantelidakis [Pa03] with the case of odd $n$ and $m\ge3619$, a pending partial claim on its claim page. The thesis is not available, and its citation is disputed: the site gives the title On the Number of Non-isomorphic Groups of the Same Order and the year 2003, while the Mathematics Genealogy Project lists an Oxford D.Phil. of 2004 titled Contributions to the Enumeration of Finite Groups. A comment in the site's discussion thread (3 September 2026), using published tables of $g(n)$ (OEIS A000001), checks the inequality for all $n\le2048$; it is a thread post and has no claim page.

Search scope: the site's problem page (last edited 26 January 2026), its discussion thread and proof-claims tab (no proof claim), the community database (no formalized statement) and the Mathematics Genealogy Project, accessed 2026-10-07.