Higman–Sims asymptotic formula

Template:Short description 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

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