Let $\ell\ge 5$ be a prime number. Let $G/\mathbb{F}_\ell$ be a (smooth, connected) reductive algebraic group. Let $G(\mathbb{F}_\ell)^+$ be the normal subgroup of $G(\mathbb{F}_\ell)$ generated by its $\ell$-Sylow subgroups. Is it true that $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is abelian?

Comments:

(1) An answer ``for sufficiently large $\ell$'' would suffice for my purpose.

(2) I seem to understand that the group $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is solvable.

Let $\ell\ge 5$ be a prime number. Let $G/\mathbb{F}_\ell$ be a (smooth, connected) reductive algebraic group. Let $G(\mathbb{F}_\ell)^+$ be the normal subgroup of $G(\mathbb{F}_\ell)$ generated by its $\ell$-Sylow subgroups. Is it true that $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is abelian?

Comments:

(1) An answer ``for sufficiently large $\ell$'' would suffice for my purpose.

(2) I seem to understand that the group $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is solvable.

Let $\ell\ge 5$ be a prime number. Let $G/\mathbb{F}_\ell$ be a (smooth, connected) reductive algebraic group. Let $G(\mathbb{F}_\ell)^+$ be the normal subgroup of $G(\mathbb{F}_\ell)$ generated by its $\ell$-Sylow subgroups. Is it true that $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is abelian?

Comments:

(1) An answer ``for sufficiently large $\ell$'' would suffice for my purpose.

(2) I seem to understand that the group $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is solvable.

Let $\ell\ge 5$ be a prime number. Let $G/\mathbb{F}_\ell$ be a (smooth, connected) reductive algebraic group. Let $G(\mathbb{F}_\ell)^+$ be the normal subgroup of $G(\mathbb{F}_\ell)$ generated by its $\ell$-Sylow subgroups. Is it true that $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is abelian?

Comments:

(1) An answer ``for sufficiently large $\ell$'' would suffice for my purpose.

(2) I seem to understand that the group $G(\mathbb{F}_\ell)/G(\mathbb{F}_\ell)^+$ is solvable.