# Generating function for groups of order n

1. Jun 6, 2012

### Number Nine

I've done some searching and have thus far come up empty handed, so I'm hoping that someone here knows something that I don't.

I'm wondering if there has been any work on the enumeration of groups of order n (up to isomorphism); specifically, has anyone derived a generating function? Ideally someone would have one for all groups of order n, but I would imagine that there must at least be one for, say, finite abelian groups?

2. Jun 6, 2012

### Number Nine

Alright; I found a sort-of-answer to one half of my question that someone may find interesting, so I'll post it here.

Theorem: Let n be a positive integer with prime factorization $\prod p_{k}^{e^{k}}$, then the number of abelian groups of order n, up to isomorphism, is given by $\prod \rho(e^{k})$, where $\rho(m)$ is the number of partitions of the integer m.

Useful note: The partition function $\rho(n)$ is horrifically complicated, and is given to us courtesy of Ramanujan. It's easier to use the following generating function...
$$\sum_{n=0}^{\infty}\rho(n)q^{n} = \prod_{j=1}^{\infty}\frac{1}{1-q^{j}}\hspace{3 mm} where\hspace{2 mm} |q^{j}| \le 1$$
EDIT: Apparently the more general case (enumerating groups of order n) is an unsolved problem, which is driving me crazy enough that I've picked up a few books on finite group theory. The problem looks to be very closely tied with the distribution of prime numbers, so this might be difficult...

Last edited: Jun 6, 2012
3. Jun 6, 2012

### Vargo

Cool fact. Thanks.