From almost tight p-group enumeration to $2$-group domination among finite groups
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 2/27m3−4/9m2+Ω(m)≤ logp f(pm)≤ 2/27m3+O(m5/2). In the same paper in 1965, Sims speculated on improving the error term (O(m8/3) at that time) in the upper bound. To show that f(pm)≤ 2/27m3+O(m2) is now known as Sims' conjecture. Our first main result proves Sims' conjecture in a strong form as follows: logp f(pm)≤ 2/27m3−4/9m2+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 ≤ n, and F2(n) be the number of $2$-groups of order ≤ n. Then limn→∞F2(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: for 1<n<2m, f(n)<f(2m). Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalisation by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a p−Ω(m2) fraction of groups of order pm have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser--Li--Qiao--Rogovskyy (\emph{arXiv}, 2025) for p-groups of Frattini class $2$ yields the linear error term.
Related
- Almost Tight Bounds for Isomorphism Testing and Basis Construction in Finite Abelian Groups
- Finite presentations of metabelian groups: effective enumeration via Laurent relations
- Finite Groups with Few Almost Maximal Subgroups
- Almost all finite groups are $2$-groups of class two
- 1-genericity and almost everywhere domination
BibTeX
@misc{li2026from,
title = {{From almost tight $p$-group enumeration to $2$-group domination among finite groups}},
author = {Yinan Li and Youming Qiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14509},
eprint = {2610.14509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14509},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From almost tight p-group enumeration to $2$-group domination among finite groups
AU - Yinan Li
AU - Youming Qiao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.14509
UR - https://arxiv.org/abs/2610.14509
AB - 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 2/27m^3−4/9m^2+Ω(m)≤ log_p f(p^m)≤ 2/27m^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)≤ 2/27m^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)≤ 2/27m^3−4/9m^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 ≤ n, and F_2(n) be the number of $2$-groups of order ≤ n. Then lim_n→∞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: for 1<n<2^m, f(n)<f(2^m). Our proof refines Sims' enumeration framework using graded Lie brackets, constraints on deeper commutator coordinates, and incidence and Fourier estimates for alternating forms. Normalisation by changes of generating bases then determines the quadratic term in the upper bound. We also prove that all but a p^−Ω(m^2) fraction of groups of order p^m have a central elementary abelian Frattini subgroup. Combining this concentration with the sharp enumeration bound of Bläser--Li--Qiao--Rogovskyy (\emph{arXiv}, 2025) for p-groups of Frattini class $2$ yields the linear error term.
ER -