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Math Help - center of SL(n,F)

  1. #1
    Member Goku's Avatar
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    center of SL(n,F)

    Show that Z(SL(n,F)) \cong {a \in F* | a^n = 1}.

    I know that if X \in Z(SL(n,F)) then det(X) = det(aI) = a^n which has to equal 1 since it is in SL(n,F).

    Any Help greatly appreciated.

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  2. #2
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    Hi!

    You write X=aI, so i suppose you know that a matrix commutes with any other matrix iff it is a multiple of the identity. The same holds for SL(n,F).

    Banach
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  3. #3
    Member Goku's Avatar
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    if X Z(SL(n,F)) then det(X) = det(aI) = a^n which has to equal 1 since it is in SL(n,F).
    So does this count as a proof.
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  4. #4
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    Goku: What you've written doesn't quite make sense. The center will \{aI_n : a^n=1\}.

    Clearly this set is contained in the center. Conversely, let A=(a_{i j}) be in the center. Let E_{i j} be the matrix with 1 in the (i,j)th position and 0 elsewhere. Then I_n+E_{i j}\in SL(n, F) if i\ne j. Now use the fact that this matrix must commute with A.
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