Higman–Sims asymptotic formula

Short description: Asymptotic estimate in group theory

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

Statement

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

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

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

References

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