# Different factorizations of p^n

1. Feb 23, 2005

### DarkEternal

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. Feb 24, 2005

### matt grime

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.