I'm new to group theory. I'm trying to find which elements of $GL_{2}(F)$ commute with all other elements of $GL_{2}(F)$. I have proved that all diagonal matrices as following $\begin{pmatrix}a & 0\\ 0 & a \end{pmatrix}$, are commute with all other elements in $GL_{2}(F)$. But how can I prove that there are no other elements that fulfill the condition?
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Which matrices commute with $\pmatrix{1&0\1&1}$? – Angina Seng Nov 10 '18 at 18:37
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See also this question; the center of $GL_2(F)$. – Dietrich Burde Nov 10 '18 at 19:06
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Assume first that $F$ is algebraically closed. Then any $g$ in $G = GL_2(F)$ is conjugate to a diagonal matrix. Any central $g$ (i.e., any $g$ that commutes with all of $G$) must thus be diagonal itself, and the fact that it commutes with any permutation matrix forces all its diagonal matrices to be equal, i.e., $g = \lambda\operatorname{id}$ for some $\lambda\in F$.
For the general case, fix some central $g\in G$. For any $x\in M_2(F)$, the sum $x + \lambda$ must lie in $GL_2(\overline{F})$ for some $\lambda\in \overline{F}$, since $\det(x + \lambda)$ is a polynomial in $\lambda$ of degree (exactly) $2$. Since $\lambda$ is central (in $GL_2(\overline{F})\supset G$), it follows that $g$ must commute with every element of $M_2(F)$. Define $f:M_2(\overline{F}) \to M_2(\overline{F})$ by $f(x) = xg - gx$. Then each entry $f_{ij}$ is a linear function of the $x_{ij}$ with coefficients in $F$, so $f$ vanishes on $M_2(F)$ only if it vanishes on $M_2(\overline{F})$. It follows that $g$ is central in $GL_2(\overline{F})\supset G$, and the same argument above applies.
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