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Thread: 2.2.7, need help

  1. #1
    Junior Member
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    2.2.7, need help

    this problem is on dummit, "abstract Algebra",
    it says
    Let $\displaystyle n\in N$, with $\displaystyle n\geq3$. Prove the following:
    (a). Z($\displaystyle D_{2n})=1$ if n is odd.

    (b). Z($\displaystyle D_{2n})={ 1,r^{k} }$ if n=2k.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by mancillaj3 View Post
    this problem is on dummit, "abstract Algebra",
    it says
    Let $\displaystyle n\in N$, with $\displaystyle n\geq3$. Prove the following:
    (a). Z($\displaystyle D_{2n})=1$ if n is odd.

    (b). Z($\displaystyle D_{2n})={ 1,r^{k} }$ if n=2k.
    According to my word processor I've spent the last 52 minutes trying to solve this by induction. Which is silly.

    I gave a short bit of thought before this, wondering if there was an easier, more subtle proof. I have just realised that there is. So here is a definition and a hint:

    We can define the group thus: $\displaystyle D_{2n} = <\alpha, \beta | \alpha^n=\beta=1, \beta \alpha \beta = \alpha^{-1}>$.

    Now, when does the equality $\displaystyle a \equiv -a \text{ mod } n$ hold, and why is this relevant here?
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