Enumeration of groups of prime power order

Enumeration of groups of prime power order refers to the problem of enumerating all finite groups of order $p^{n}$ where $p$ is a prime and $n$ is a positive integer.

The problem was first studied by Higman and Sims. They introduced the function:

$f(n,p)$

which returns the number of groups of order $p^{n}$. The following facts are known about $f(n,p)$:

• $f(1,p)=1$ for all primes $p$
• $f(2,p)=2$ for all primes $p$
• $f(3,p)=5$ for all primes $p$
• $f(4,2)=14$ and $f(4,p)=15$ for any odd prime $p$

For higher $n$, $f(n,p)$ has not yet been found to have a closed expression. Higman proved the following estimate:

$f(n,p)=p^{An^{3}}$

where $A$ depends on $n$ and $p$, and we have:

${\frac {2}{27}}-\varepsilon _{n}\leq A\leq {\frac {2}{27}}+\varepsilon _{n}$

with $\varepsilon _{n}$ independent of $p$, and approaching zero as $n\to \infty$. In other words, we can loosely say:

$f(n,p)\simeq p^{2n^{3}/27}$

See also

• Group number function

References

• Enumerating p-groups by Graham Higman, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), (Year 1959): ^{More info}
• Paper:Simsenumpgrp^{More info}