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Math Help - Isomorphism Question

  1. #1
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    Isomorphism Question

    Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to
    D(n/2).

    Last edited by wutang; May 19th 2010 at 06:23 PM.
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  2. #2
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    Quote Originally Posted by wutang View Post
    Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to D(n/2).


    Hints: if D_{2n}=\left\{a,b\;;\;a^2=b^n=1\,,\,aba=b^{-1}=b^{n-1}\right\} , then:

    1) Z\left(D_{2n}\right)=\{1,b^{n/2}\}

    2) D_{2n}/Z\left(D_{2n}\right)=\left\{\overline{a}\,,\,\over  line{b}\;;\;\overline{a}^2=\overline{b}^{n/2}=\overline{1}\,,\,\overline{a}\overline{b}\overl  ine{a}=\overline{b}^{-1}\right\} , with \overline{x}:=xZ\left(D_{2n}\right)\in D_{2n}/Z\left(D_{2n}\right)\,,\,x\in D_{2n}

    Tonio
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  3. #3
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    It should say that Dn/Z(Dn) is isomorphic to D(n/2). I understand your definition for Dn/Z(Dn), but I don't get how to set up the isomorphism. I think I should use the fact that any group generated by a pair of elements of order 2 is dihedral to get the isomorphism from Dn/Z(Dn) to D(n/2) ?
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  4. #4
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    never mind, I think I just have to play with the elements in Dn/Z(Dn) until I get it to look that D(n/2).
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