The Details: Let $G$ be a group and let $X\subseteq G$. Let $n\in\Bbb N$. Recall that for $H\le G$, $${\rm conj}_H(X)=\{ hxh^{-1}\mid x\in X, h\in H\}.$$ Define $$B_X(n)=\bigcup_{k=0}^n\underbrace{{\rm conj}_G(X)^{\pm 1}\dots{\rm conj}_G(X)^{\pm 1}}_{k\text{ times.}} $$ Note that the idea here is...

This is Exercise 1.8.4(1) of Springer's, "Linear Algebraic Groups (Second Edition)". It is not a duplicate of The dimension of $\mathbb P^n$ is $n$ because I'm after a particular perspective; namely, the approach Springer takes (using transcendence degree and not Krull dimension). The Question: ...