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My question is to find the center of GL(2,F) and SL(2,F) where $F=\mathbb{Z},\mathbb{C}$

My attempt:

Generalising the identity element, if we take $\pmatrix{ a& 0\\\ 0& a}$. Then these elements commute with every 2x2 matrix irrespective of which $F$ we take.

So is it safe to say that these are in center of all these groups?

Thibaut Dumont
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Foggy
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  • Yes for $F=\mathbb{C}$, but this needs a proof :). – Thibaut Dumont Aug 19 '15 at 07:20
  • even if F=Z then too this will be center – Foggy Aug 19 '15 at 07:24
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    Sure, but that's just restating the definition of the center, and you also haven't proved that every central element is of that form. – anomaly Aug 19 '15 at 07:25
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    @Foggy : However you need to be careful that $\pmatrix{2&0\0&2}$ is not in $GL_2(\mathbb{Z})$ but does commute with all the matrices of the latter. – Thibaut Dumont Aug 19 '15 at 07:26
  • @ThibautDumont why not – Foggy Aug 19 '15 at 07:29
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    $GL_2(\mathbb{Z})$ is defined to be the set of matrices $g$ with integer coefficients such that $det(g)\in \mathbb{Z}^*={\pm 1}$. This is to ensure that the inverse is again in $GL_2(\mathbb{Z})$. But the inverse of $diag(2,2)$ is $diag(1/2,1/2)$. – Thibaut Dumont Aug 19 '15 at 07:32

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