# Classify commutative groups

• March 4th 2010, 07:00 PM
brisbane
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.
• March 4th 2010, 07:06 PM
tonio
Quote:

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.

$1,000=2^3\cdot 5^3\Longrightarrow$ 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