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Math Help - center, centralizer

  1. #1
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    center, centralizer

    Let D4 = {e, r, r2, r3, f, fr, fr2, fr3}, where r4 = f2 = e and rf = fr−1 = fr3.
    (a) Find the centralizer CD4(r) of r and the centralizer CD4(f) of f in D4.
    (b) Find the center Z(D4) of D4.


    The centralize is C(a) with all elements of D4 that commute with a.
    We want xr=rx
    C(r)={e,r,r^2,r^3,f}
    want xf=fx
    C(f)={e,r,f,fr^2}



    center is elements that commute with every other element of D4 z(D4)
    want xg=gx
    Z(D4)={e,r,f}

    not sure if any of the values are correct or if I'm missing some.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kathrynmath View Post
    Let D4 = {e, r, r2, r3, f, fr, fr2, fr3}, where r4 = f2 = e and rf = fr−1 = fr3.
    (a) Find the centralizer CD4(r) of r and the centralizer CD4(f) of f in D4.
    (b) Find the center Z(D4) of D4.


    The centralize is C(a) with all elements of D4 that commute with a.
    We want xr=rx
    C(r)={e,r,r^2,r^3,f}
    want xf=fx
    C(f)={e,r,f,fr^2}


    center is elements that commute with every other element of D4 z(D4)
    want xg=gx
    Z(D4)={e,r,f}

    not sure if any of the values are correct or if I'm missing some.
    I'm fairly sure that the rule is that if n>2 and 2\mid n that, with your notation, \mathcal{Z}\left(D_{2n}\right)=\left\langle r^{\frac{n}{2}}\right\rangle. So, I'd say that you're answer is probably wrong.


    That said, you haven't shown any work we could judge?
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