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Math Help - Classify commutative groups

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by brisbane View Post
    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
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