Higman-Sims asymptotic formula on number of groups of prime power order

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Statement

Quick version

Let p be a (fixed) prime number. Define f(n,p) as the number of groups of order p^n (see also group of prime power order, number of groups of given order). Then:

f(n,p) = p^{(2/27 + o(1))n^3}

Full version

Let p be a (fixed) prime number. Define f(n,p) as the number of groups of order p^n (see also group of prime power order, number of groups of given order). Define:

A(n,p) := \frac{\log f(n,p)}{n^3 \log p}

Then:

A(n,p) = \frac{2}{27} + O(n^{-1/3})

Note that the big-O notation is with respect to n (sending n \to \infty and not with respect to p). However, the constants under the big-O notation may depend on p.