Exercise 14.3.14

Proof. (a) To build an $2\times 2$ matrix $A\in \mathrm{GL}(2,{𝔽}_q)$, we must first choose a nonzero vector $u\in {𝔽}_q^2$ (the first column of the matrix), then a vector $v\notin {𝔽}_{q}u$ (the second column). Since $|{𝔽}_q^2\setminus \{0\}|=q^2-1$, and $|{𝔽}_q^2\setminus {𝔽}_{q}u|=q^2-q$, we obtain

(b) $\mathrm{SL}(2,{𝔽}_q)$ is the kernel of the surjective homomorphism $\mathit{det}:\mathrm{GL}(2,{𝔽}_q)\to {𝔽}_q^{\text{∗}}$, so that

Therefore $|\mathrm{GL}(2,{𝔽}_q)|∕|\mathrm{SL}(2,{𝔽}_q)|=|{𝔽}_q^{\text{∗}}|$, thus

The definition

gives

(c) Consider the restriction of the canonical projection $\pi :\mathrm{GL}(2,{𝔽}_q)\to \mathrm{GL}(2,{𝔽}_q)∕{𝔽}_q^{\text{∗}}{I}_n$ to $\mathrm{SL}(2,{𝔽}_q)$. We obtain

If $A\in \mathrm{SL}(2,{𝔽}_q)$ is such that $[A]=[I_2]$, then $A=\lambda I_2,\lambda \in {𝔽}_q^{\text{∗}}$, and $\det <!--\; FUNCTION\; APPLICATION\; -->(A)={\lambda }^2=1$, so that $\lambda =\pm 1$. We have proved $\ker <!--\; FUNCTION\; APPLICATION\; -->({\pi }_1)=\{I_2,-I_2\}$ (where $I_2=-I_2$ if the characteristic is 2). Therefore

Note that $\phi$ is not surjective.

By the definition given in the mathematical notes, $\mathrm{Im}\phi =\mathrm{PSL}(2,{𝔽}_q)$, thus

If $q=2^n$, then the characteristic of ${𝔽}_q$ is 2, and $I_2=-I_2$, so that $|\mathrm{PGL}(2,{𝔽}_q)|=|\mathrm{SL}(2,{𝔽}_q)|$. Otherwise $|\mathrm{PGL}(2,{𝔽}_q)|=\frac{1}{2}|\mathrm{SL}(2,{𝔽}_q)|$. This proves

(d)

(e) The same reasoning as in part (a) shows that

thus, for $q=2$,

As in parts (b) and (c),

For $q=2$, $I_2=-I_2$, and ${𝔽}_q^{\text{∗}}=\{1\}$, thus

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