Every finite Abelian group can be written as the direct sum of cyclic groups of prime power order.
To find the number of Abelian groups of order 2025 we have to find the number of ways to partition the factors of 2025 into prime powers.
Now 2025 = 34 × 52.
We can partition 34 into prime powers 5 ways:
- 34
- 33 × 3
- 32 × 32
- 32 × 3 × 3
- 3 × 3 × 3 × 3
And we can partition 52 two ways:
- 52
- 5 × 5
A couple of notes here. First, we only consider positive powers. Second, two partitions are considered the same if they consist of the same factors in a different order. For example, 3 × 3 × 32 and 32 × 3 × 3 are considered to be the same partition.
It follows that we can partition 2025 into prime powers 10 ways: we choose one of the five ways to partition 34 and one of the two ways to partition 52. Here are all the Abelian groups of order 2025:
- ℤ81 ⊕ ℤ25
- ℤ81 ⊕ ℤ5 ⊕ ℤ5
- ℤ27 ⊕ ℤ3 ⊕ ℤ25
- ℤ27 ⊕ ℤ3 ⊕ ℤ5 ⊕ ℤ5
- ℤ9 ⊕ ℤ9 ⊕ ℤ25
- ℤ9 ⊕ ℤ9 ⊕ ℤ5 ⊕ ℤ5
- ℤ9 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ25
- ℤ9 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ5 ⊕ ℤ5
- ℤ3 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ25
- ℤ3 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ3 ⊕ ℤ5 ⊕ ℤ5
Given a prime number q, there are as many ways to partition qk as the product of positive prime powers as there are ways to partition k into the sum of positive integers, denoted p(k). What we have shown above is that the number of Abelian groups of order 34 52 equals p(4) p(2).
In general, to find the number of Abelian groups of order n, factor n into prime powers, then multiply the partition numbers of the exponents in the factorization.