p.7 Paragraph following 1.3. Add: The morphism $e\to G$ factors through $i\circ m_H\circ (id_H,inv_H)\colon H\to G$ and provides a neutral element for $H$.
p.15 In the proof of Proposition 1.29, G should be X (3 times). (Long Liu)
p.51 First paragraph of Section 2.g. The map (of abstract groups) $G(k^{\mathrm{s}% })\rightarrow\pi_{0}(G_{k^{s}})$ need not be surjective. When $k$ is not perfect, it is better to define the étale group scheme $\pi_{0}(G)$ by applying the functor $\pi_{0}$ to $(G,m,e,inv)$. See here (Alex Youcis).
p.89 In the proof of 4.14, the map $e_j\mapsto T_{ij}$ should be $e_j\mapsto T_{ji}$.
p.108 In 5.40, the second $q$ should be $q'$.
p.160 After the first paragraph, $A$ is an abelian variety
(Haohao Liu).
Replace
(a) $\mathrm{Ext}(A,\mathbb{G}_m)\simeq H^1(A,\mathcal{O}_A^\times)$
with (a) $\mathrm{Ext}(A,\mathbb{G}_m)\simeq$ the set of
primitive elements in $H^1(A,\mathcal{O}_A^\times)$ (Long Liu).
p.234 In the last sentence of Example 12.10, the base field is algebraically closed, otherwise an extension of diagonalizable groups need not be diagonalizable, even if commutative.
Section 13c, p.260. From an email: In Chapter 13, Tori Acting on Schemes, about the proof of Proposition 13.22, it is not clear for me its relationship with the commutative algebra statement in 13.27. To me it seems that 13.27 corresponds to the following statement: If x is a fixed point, regular on both X and Z, then x is regular on X(Z). [I need to look at this.]
p.321 Need to assume the groups are commutative in Theorem 15.37. Without that the second row of the second diagram in the proof need not be exact (the pushout is more complicated). The Frobenius kernel of $\text{SL}_2$ is a counterexample to the theorem. The result is only used in the proof of 16.13, where the groups are commutative. (Damian Sercombe)
p. 322 The statement of Theorem 15.39 is correct although (apparently) the proof assumes that $G'$ is connected (mo498313)
p.412 Lemma 20.15 does not assume that $G$ is reductive, while the proof uses Proposition, 17.61 which does. Here is a possible workaround: consider the reductive quotient $G'= G/R_u(G)$. Let $Z'$ be the image of $Z(\lambda)$ and $T'$ be the image of $T$ in $G'$. Then $Z'$ is included in the centralizer of $T'$, and hence is included in $T'$ because Prop 17.61 applies to $G'$. Also we have an extension $1\rightarrow Z(\lambda)\cap R_u(G)\rightarrow Z(\lambda)\rightarrow Z'\rightarrow 1$ in which the kernel and quotient are solvable; hence $Z(\lambda)$ is solvable and this is enough to carry on with the proof: $P(\lambda)$ is solvable etc. Matthieu Romagny
p.440, 21.19. Should read "Recall (14.60)" rather than "Recall (14.59)". Matthieu Romagny
p.457, line 2 $i\alpha+i\beta$ should be $i\alpha+j\beta$ (obviously).
p.465, 22.8 The statement $X\subset X_0+P(\Phi)$ is only true after you mod out by $X_0$, as can be seen already for GL$_2$ (Syed Waqar Ali Shah)
p.478, proof of 22.42. The $V_i$ in the first decomposition should be the eigenspaces for the action of $Z$.
p.534, line 11 The term $X^tSX^{-1}$ should read $S^{-1}X^tS$. (Friedrich Knop)
p.562. In the statements of Theorems 25.62 and 25.63, the center of the simply connected cover should be replaced by the kernel of the isogeny (Dylan Chow)
p.563. Section g. I sometimes write "cohomology group" for "cohomology set" ($G$ is usually noncommutative) (Dylan Chow).
Appendix A. For a sheaf of $k$-algebras $\mathcal{O}_X$, I am implicitly allowing the $k$-algebra $\mathcal{O}_X(U)$ to be not finitely generated for some open $U$, contrary to my convention p.569.