Year: 2026
Type: preprint
Abstract: Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. The pioneering works of Higman (\emph{Proc. Lond. Math. Soc.}, 1960) and Sims (\emph{Proc. Lond. Math. Soc.}, 1965), with improvement by Newman and Seeley, establish that \[ \frac{2}{27}m^3-\frac{4}{9}m^2+Ω(m)\leq \log_p f(p^m)\leq \frac{2}{27}m^3+O(m^{5/2}). \] In the same paper in 1965, Sims speculated on improving the error term ($O(m^{8/3})$ at that time) in the upper bound. To show that $f(p^m)\leq \frac{2}{27}m^3+O(m^2)$ is now known as Sims' conjecture. Our first main result proves Sims' conjecture in a strong form as follows: \[ \log_p f(p^m)\leq \frac{2}{27}m^3-\frac{4}{9}m^2+O(m). \] Techniques developed to prove the above lead to progress on some well-known open questions in group enumeration in affirmative, including: (1) 2-group domination in finite group enumeration (Pyber, \emph{Eur. Congr. Math.}, 1996): Let $F(n)$ be the number of groups of order $\leq n$, and $F_2(n)$ be the number of $2$-groups of order $\leq n$. Then $$ \lim_{n\to\infty}\frac{F_2(n)}{F(n)} =1. $$ (2) A question attributed to Higman and Erdős (see Blackburn--Neumann--Venkataraman, \emph{Cambridge Tracts in Maths.}, 2007) when $m$ is large enough: $$ \text{for } 1
Source: arXiv (Cornell University)
Authors: Li, Yinan, Qiao, Youming
Institutions: Wuhan University, University of Technology Sydney
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221750044