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

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

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. pp. 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.