Let $G$ be a diagonalizable algebraic group over a field $K$ of characteristic $p > 0$. Let $X$ be the character group of $G$ (algebraic group homomorphisms $G \to K^\times$). We know $X$ is finitely generated abelian.
Claim: $X$ is $p$-torsion free.
If $\chi \in X$ with $\chi^p = 1$, then $\chi(x)^p = 1$ so the image of $\chi$ lands in the subgroup of $p$th roots of unity of $K^\times$. In the notes I am reading, they immediately conclude that $\chi(x) = 1$ for all $x \in G$, but I don't see how this follows.
Since $K$ has characteristic $p$, it has no nontrivial $p$th roots of unity. The polynomial $x^p-1$ factors as $(x-1)^p$ so the only $p$th root of unity is $1$.