Limit to Infinity of Number of p-Groups of Order p^m

Theorem

Let $p$ be a prime number.

Let $m \in \N$ be a natural number

Let $\map \nu {p^n}$ denote the $\nu$ function of $p^n$: the number of group types of order $p^m$.

Then:

$\map \nu {p^m} = p^{A m^3}$

where:

$\ds \lim_{m \mathop \to \infty} A = \dfrac 2 {27}$

Proof

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Sources

• 1960: Graham Higman: Enumerating $p$-Groups. $\text I$: Inequalities (Proc. London Math. Soc. Ser. 3 Vol. 10, no. 1: pp. 24 – 30)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,07407407407 \ldots$