Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Generating function for groups of order n

  1. Jun 6, 2012 #1
    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. jcsd
  3. Jun 6, 2012 #2
    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 [itex]\prod p_{k}^{e^{k}}[/itex], then the number of abelian groups of order n, up to isomorphism, is given by [itex]\prod \rho(e^{k})[/itex], where [itex]\rho(m)[/itex] is the number of partitions of the integer m.

    Useful note: The partition function [itex]\rho(n)[/itex] is horrifically complicated, and is given to us courtesy of Ramanujan. It's easier to use the following generating function...
    [tex]\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[/tex]
    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
  4. Jun 6, 2012 #3
    Cool fact. Thanks.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Generating function groups Date
Generalization of combinatorial generating functions? Dec 31, 2014
Function to generate linearly independent vectors Nov 8, 2012
Generating function Dec 27, 2011
Generating function Dec 22, 2011
Dirichlet Generating function and Poles. Jul 13, 2007