Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A\_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.


Let $X(G)\_{\mathbb{Q}}$ be the addtive group of homomorphisms $\chi: G \to GL(1)$ that are defiend over $\mathbb{Q}$.


Then I am wondering why $X(G)\_{\mathbb{Q}}$ is a free abelian group of rank $k$. I know such $\chi$ factor through $G/[G,G]$. But how can we relate it with largest central subgroup $A\_G$?



Does every character from group factor through largest central subgroup?
Every connected reductive group $G$ is a product $Z\cdot G'$ where $Z$ is the connected center of $G$ and $G'$ is the derived group of $G$; moreover $Z\cap G'$ is finite. This is a standard result. See for example 19.25 or 21.61 of Milne, Algebraic Groups, CUP 2017.


