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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.
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