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

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

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

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.