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Math Help - Finite Group Property

  1. #1
    Rhymes with Orange Chris L T521's Avatar
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    Finite Group Property

    I'm helping a friend with a problem that I can't figure out. I would appreciate some guide lines on how to tackle this:

    Prove: Every finite group is a disjoint union of cyclic subgroups.

    If a group is Abelian, this should be a restatement of the Fundamental Theory of Finite Abelian Groups. I also know that the dihedral groups D_n satisfies the requirements for this statement. But how do we tackle the case when we have a general group G that is not abelian?

    I would appreciate any help that you're able to give.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    I'm helping a friend with a problem that I can't figure out. I would appreciate some guide lines on how to tackle this:

    Prove: Every finite group is a disjoint union of cyclic subgroups.

    If a group is Abelian, this should be a restatement of the Fundamental Theory of Finite Abelian Groups. I also know that the dihedral groups D_n satisfies the requirements for this statement. But how do we tackle the case when we have a general group G that is not abelian?

    I would appreciate any help that you're able to give.
    Do you mean product? If you have two groups G,G' how do you define an operation on G\cup G'?
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  3. #3
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    Quote Originally Posted by Chris L T521 View Post
    I'm helping a friend with a problem that I can't figure out. I would appreciate some guide lines on how to tackle this:

    Prove: Every finite group is a disjoint union of cyclic subgroups.

    If a group is Abelian, this should be a restatement of the Fundamental Theory of Finite Abelian Groups. I also know that the dihedral groups D_n satisfies the requirements for this statement. But how do we tackle the case when we have a general group G that is not abelian?

    I would appreciate any help that you're able to give.

    I'm not sure I'm understanding: what do you mean "disjoint union" of subgroups? At least they all have the unit as acommon element...Perhaps you're thinking of direct product? Because if you are then the claim is false.

    Tonio
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