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Claim. Let $g(n)$ be the number of groups of order $n$. For every odd $n$ and every $m\ge3619$ with $n\le2^m$, $g(n)\le g(2^m)$. The site credits this to Pantelidakis's Oxford DPhil thesis, which it cites as [Pa03], On the Number of Non-isomorphic Groups of the Same Order (2003); its curator found the reference in Blackburn, Neumann and Venkataraman's Enumeration of Finite Groups [BNV07]. The Mathematics Genealogy Project lists Ioannis Pantelidakis's Oxford D.Phil. as Contributions to the Enumeration of Finite Groups (2004), supervised by P. M. Neumann, so the thesis's title and year are unsettled. This page is named by the year of the citation the site credits, filled to its first day; the thesis itself, which would settle the date, is not available. The claim bears on Problem 1160.

Covers. Odd $n$ with $m\ge3619$. Not even $n$, and not the finitely many odd cases with $m<3619$, to which, as the curator notes, the result reduces the odd case.

Depends on. No page of this wiki.

Standing. Claimed: a thesis, not a journal publication, under the site's label OPEN. The curator wrote in the site's discussion thread on 26 January 2026 that he had found no online copy of the thesis to check the result.