Thread: center of SL(n,F)

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

I know that 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).

Any Help greatly appreciated.

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

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.

Goku: What you've written doesn't quite make sense. The center will .

Clearly this set is contained in the center. Conversely, let be in the center. Let be the matrix with 1 in the th position and 0 elsewhere. Then if . Now use the fact that this matrix must commute with .