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Math Help - center of general linear group

  1. #1
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    center of general linear group

    I am trying to show that Z(GL(n,F*)) \cap SL(n,F*) \cong T(n) ... This is for any non-zero Field

    Where Z is the center of General Linear Group.
    and T(n)={a^n=1 | a \in F*}

    Im trying to figure this out first Z(GL(n,F*)) SL(n,F*), since the determinant in this set is suppose to be 1, then det(aI) = det(a)det(I) = det(a), this has to be 1 to be in the set therefore a=1, which implies this set contains I(n,F*) (The identity matrix) only.

    Am I right so far???

    Don't really know what else to do.
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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
    I am trying to show that Z(GL(n,F*)) \cap SL(n,F*) \cong T(n) ... This is for any non-zero Field

    Where Z is the center of General Linear Group.
    and T(n)={a^n=1 | a \in F*}

    Im trying to figure this out first Z(GL(n,F*)) SL(n,F*), since the determinant in this set is suppose to be 1, then det(aI) = det(a)det(I) = det(a), this has to be 1 to be in the set therefore a=1, which implies this set contains I(n,F*) (The identity matrix) only.

    Am I right so far???

    Don't really know what else to do.

    You really need to show that Z:=Z\left(GL(n,\mathbb{F})\right)=\{aI\,|\,a\in\ma  thbb{F}^*\}, and then

    all falls in place.

    You can show the above by taking B\in Z and checking what the coefficients of B must be by

    analyzing BE_{ij}=E_{ij}B\,,\,\,with\,\,E_{ij}= the matrix with 1 in the ij entry and zero everywhere else.

    Tonio

    Pd Pay attention to the notation of the general linear group: it is not F* in it but F...
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