Higman–Sims asymptotic formula

In finite group theory, the Higman–Sims asymptotic formula gives an asymptotic estimate on number of groups of prime power order.

Statement

Let ${\displaystyle p}$ be a (fixed) prime number. Define ${\displaystyle f(n,p)}$ as the number of isomorphism classes of groups of order ${\displaystyle p^{n}}$. Then:

${\displaystyle f(n,p)=p^{{\frac {2}{27}}n^{3}+{\mathcal {O}}(n^{8/3})}}$

Here, the big-O notation is with respect to ${\displaystyle n}$, not with respect to ${\displaystyle p}$ (the constant under the big-O notation may depend on ${\displaystyle p}$).

References

• Kantor, William M. (1990). "Some topics in asymptotic group theory". Groups, Combinatorics and Geometry. Durham. pp. 403–421.
• Higman, Graham (1960). "Enumerating p-Groups. I: Inequalities". Proceedings of the London Mathematical Society. 3 (1): 24–30. doi:10.1112/plms/s3-10.1.24.
• Sims, Charles C. (1965). "Enumerating p-Groups". Proceedings of the London Mathematical Society. 3 (1): 151–166. doi:10.1112/plms/s3-15.1.151.