# Thread: Classify commutative groups

1. ## Classify commutative groups

The following is a review problem for a midterm I have:

Classify all commutative groups with 1,000 elements up to isomorphism. How many isomorphism classes of such groups are there?

If anyone can help me out explicitly, that'd be great -- it's not a problem I need to solve, but rather one I need to understand conceptually.

2. Originally Posted by brisbane
The following is a review problem for a midterm I have:

Classify all commutative groups with 1,000 elements up to isomorphism. How many isomorphism classes of such groups are there?

If anyone can help me out explicitly, that'd be great -- it's not a problem I need to solve, but rather one I need to understand conceptually.

$\displaystyle 1,000=2^3\cdot 5^3\Longrightarrow$ the number of abelian groups of order 1000 up to isomorphism equals the number of partitions of 3 times the number of partitions of 3 = 9 .

This follows at once from the Fundamental Theorem of finitely generated abelian groups.

Tonio