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Math Help - Dihedral Group D3 and Direct Products

  1. #1
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    Dihedral Group D3 and Direct Products

    Hey there everyone,

    Does D3 (The dihedral group of order 6) is indecomposable?
    Proof is needed...


    TNX!
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by WannaBe View Post
    Hey there everyone,

    Does D3 (The dihedral group of order 6) is indecomposable?
    Proof is needed...


    TNX!
    If it was, what would it be decomposed into? What are the orders of these groups that it would be decomposed into? Assume it was decomposable into these two groups. Then dos this gives you a contradiction? (Hint: look at abelian-ness)...
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  3. #3
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    I tried to figure out what is the contradiction we get from assuming that it's decomposable...I first thought about normality...If D3 is simple, then it's not decomposable, as required... The proof is kind of a Lemma for another proof, so I would be delighted if you'll be able to be more specific...The orders of the groups it would be decomposed to would be 3&2 (or 6 and 1 but it's trivial)... D3 has a sub-group of order 2 and a sub-group of order 3...So I can't realy see the contradiction...

    Tnx in advance...
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by WannaBe View Post
    I tried to figure out what is the contradiction we get from assuming that it's decomposable...I first thought about normality...If D3 is simple, then it's not decomposable, as required... The proof is kind of a Lemma for another proof, so I would be delighted if you'll be able to be more specific...The orders of the groups it would be decomposed to would be 3&2 (or 6 and 1 but it's trivial)... D3 has a sub-group of order 2 and a sub-group of order 3...So I can't realy see the contradiction...

    Tnx in advance...
    Think about commutativity in your group. D_3 is not commutative. What about your direct product?
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  5. #5
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    Well... D3 isn't commutative indeed... Each subgroup of order 2 of D3 is of the form
    {1,a} where a is a reflection and it's commutative...
    I can't think of an example of a subgroup of order 3 so I can't answer your question completely....If the subgroup of order 3 of D3 is also not commutative, so because of the isomorphism between the two, we will get a contradiction..But what are the subgroups of order 3 of D3?

    TNX
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by WannaBe View Post
    Well... D3 isn't commutative indeed... Each subgroup of order 2 of D3 is of the form
    {1,a} where a is a reflection and it's commutative...
    I can't think of an example of a subgroup of order 3 so I can't answer your question completely....If the subgroup of order 3 of D3 is also not commutative, so because of the isomorphism between the two, we will get a contradiction..But what are the subgroups of order 3 of D3?

    TNX
    As you said, D_3 is non-commutative. However, if it was decomposable when what would it look like? It would have to be the direct product of C_2 and C_3, the cyclic groups of order 2 and 3 respectively.

    Is this group abelian?
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  7. #7
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    This group is abelian indeed....In contradiction to the fact that D3 isn't abelian... Tnx a lot man!
    I'm pretty sure D3 has a semidirect deomposite...Am I right??

    TNX
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  8. #8
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    Quote Originally Posted by WannaBe View Post
    This group is abelian indeed....In contradiction to the fact that D3 isn't abelian... Tnx a lot man!
    I'm pretty sure D3 has a semidirect deomposite...Am I right??

    TNX

    Yes: D_3 has a normal sbgp. of index 2, so it's the semidirect product of this sbgp. by any of its sbgps. of order 2.

    Tonio
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  9. #9
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    Tnx a lot to you both!
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