Almost all finite groups are $2$-groups of class two
Abstract
We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most x are counted up to isomorphism and x→∞. More precisely, uniformly for 2m≤ x<2m+1, all but an O(2−cm2) proportion have order 2m and class two, for some absolute constant c>0. For each fixed prime p, we give a formula for the number of groups of order pm, with relative error O(p−m/3) as m→∞. This refines the classical logarithmic estimates for the number of p-groups. For every fixed prime p and sufficiently large m, we also prove that all but a p−m2/300 proportion of groups of order pm have central elementary abelian Frattini subgroup. For fixed p, the distribution of the minimum number of generators among groups of order pm is asymptotically supported on one or two adjacent values, with explicit probabilities depending on m modulo three.
Related
- Finite Groups with Few Almost Maximal Subgroups
- Almost Tight Bounds for Isomorphism Testing and Basis Construction in Finite Abelian Groups
- Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
- Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions
- On finite simple groups possessing a conjugacy class of size 3p^n
BibTeX
@misc{su2026almost,
title = {{Almost all finite groups are $2$-groups of class two}},
author = {Ning Su and Binzhou Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11092},
eprint = {2610.11092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11092},
}
RIS
TY - UNPB
TI - Almost all finite groups are $2$-groups of class two
AU - Ning Su
AU - Binzhou Xia
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11092
UR - https://arxiv.org/abs/2610.11092
AB - We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most x are counted up to isomorphism and x→∞. More precisely, uniformly for 2^m≤ x<2^m+1, all but an O(2^−cm^2) proportion have order 2^m and class two, for some absolute constant c>0. For each fixed prime p, we give a formula for the number of groups of order p^m, with relative error O(p^−m/3) as m→∞. This refines the classical logarithmic estimates for the number of p-groups. For every fixed prime p and sufficiently large m, we also prove that all but a p^−m^2/300 proportion of groups of order p^m have central elementary abelian Frattini subgroup. For fixed p, the distribution of the minimum number of generators among groups of order p^m is asymptotically supported on one or two adjacent values, with explicit probabilities depending on m modulo three.
ER -