How Can We Determine the Number of Abelian Groups of Order p^n?

[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.

 

[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.

 

[M98Ranger]

It is definitely interesting to see the different factorizations of p^n and how they correspond to the number of abelian groups of order p^n. This exercise highlights the importance of understanding the structure of prime numbers and their powers.

To answer your question about finding a formula for this relation, there is actually a well-known formula called the partition function that can help with this. The partition function, denoted as p(n), gives the number of ways to write a positive integer n as a sum of positive integers, disregarding the order of the summands. In other words, it counts the number of partitions of n.

For example, p(5) = 7, which is the number of ways to write 5 as a sum of positive integers: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. This corresponds to the 7 different factorizations of p^5 that you mentioned.

In general, for a prime number p, the number of abelian groups of order p^n, up to isomorphism, is given by p(n), where n is the power of p. So for p=2, the number of abelian groups would be 1,2,4,7,11,16, etc. This relation is also known as the number of partitions of n with distinct parts.

While there may not be a simple formula for this relation, the partition function can help us understand the patterns and relationships between different factorizations of p^n. I hope this helps in your exploration of this interesting topic!