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Different factorizations of p^n

  1. Feb 23, 2005 #1
    I was doing an exercise for my algebra class dealing with the number of abelian groups of order p^n, where p is a prime number, up to isomorphism. i had to fill out a table for n = 2 to n = 8. for example, for n = 2,3,4,5,6,7,8, the number of groups = 2,3,5,7,11,15,22, respectively. essentially, the different ways one can write p^n. for example, for p^5, one can write p^5, p^4 p, p^3 p p, etc... although not part of the question, i was wondering if there was an easy way to work out a formula for this relation given any n? seems interesting, but i haven't come across it yet.
     
  2. jcsd
  3. Feb 24, 2005 #2

    matt grime

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    Yes, these are partition numbers of some kind: the number of ways of writing n as the sum of integers.

    2=1+1=0+2
    3=1+1+1=1+2=0+3
    4=1+1+1+1=1+1+2=2+2=1+3=0+4

    If you think about it an abelian p group is specified by the number of subgroups of order p, p squared, p cubed and so on, and it becomes clear where the numbers come from.
     
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