Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius endomorphism, where $q = p^e$ for some natural number $e$. Assume that both $G$ and $H$ are $\sigma$-stable.

A classic result by Rosenlicht (see Corollary 16.5 in Borel), says that the canonical map $\pi:G_\sigma \rightarrow (G/H)_\sigma$ is surjective (here $X_\sigma$ denotes the $\sigma$-fixed points of $X$).

I am interested in getting a lower bound for $\frac{|G_\sigma|}{|H_\sigma|}$ as a function of $q$, when $H$ is a proper subgroup of $G$. Is it perhaps the case that $\frac{|G_\sigma|}{|H_\sigma|} \geq (q-1)$? If it's of any help, I'd be happy with the assumption $\dim G = \dim H+1$.

If such a result cannot be achieved, is there maybe a lower bound on $q$ that would allow to conclude that $\frac{|G_\sigma|}{|H_\sigma|} \geq 7$?

Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius endomorphism, where $q = p^e$ for some natural number $e$. Assume that both $G$ and $H$ are $\sigma$-stable.

A classic result by Rosenlicht (see Corollary 16.5 in Borel), says that the canonical map $\pi:G_\sigma \rightarrow (G/H)_\sigma$ is surjective (here $X_\sigma$ denotes the $\sigma$-fixed points of $X$).

When $H$ is a proper subgroup of $G$, is it the case that $\frac{|G_\sigma|}{|H_\sigma|} \geq (q-1)$?

Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius endomorphism, where $q = p^e$ for some natural number $e$. Assume that both $G$ and $H$ are $\sigma$-stable.

A classic result by Rosenlicht (see Corollary 16.5 in Borel), says that the canonical map $\pi:G_\sigma \rightarrow (G/H)_\sigma$ is surjective (here $X_\sigma$ denotes the fixed points of $\sigma$).

I am interested in getting a lower bound for $\frac{|G_\sigma|}{|H_\sigma|}$ as a function of $q$, when $H$ is a proper subgroup of $G$. Is it perhaps the case that $\frac{|G_\sigma|}{|H_\sigma|} \geq (q-1)$? If it's of any help, I'd be happy with the assumption $\dim G = \dim H+1$.

If such a result cannot be achieved, is there maybe a lower bound on $q$ that would allow to conclude that $\frac{|G_\sigma|}{|H_\sigma|} \geq 7$?

Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius endomorphism, where $q = p^e$ for some natural number $e$. Assume that both $G$ and $H$ are $\sigma$-stable.

A classic result by Rosenlicht (see Corollary 16.5 in Borel), says that the canonical map $\pi:G_\sigma \rightarrow (G/H)_\sigma$ is surjective (here $X_\sigma$ denotes the $\sigma$-fixed points of $X$).

I am interested in getting a lower bound for $\frac{|G_\sigma|}{|H_\sigma|}$ as a function of $q$, when $H$ is a proper subgroup of $G$. Is it perhaps the case that $\frac{|G_\sigma|}{|H_\sigma|} \geq (q-1)$? If it's of any help, I'd be happy with the assumption $\dim G = \dim H+1$.

If such a result cannot be achieved, is there maybe a lower bound on $q$ that would allow to conclude that $\frac{|G_\sigma|}{|H_\sigma|} \geq 7$?