Hi, I need some help with this question.

Let $\displaystyle F=F_p$ be a field and $\displaystyle V=F_p^n$.

a. The numbers of bases of $\displaystyle V$ for $\displaystyle n=2$ is equal to the order of $\displaystyle GL_2 (F)$

b. Prove the order of $\displaystyle GL_2 (F)$ is $\displaystyle p(p+1)(p-1)^2$.

c. Prove the order of $\displaystyle SL_2 (F)$ is $\displaystyle p(p+1)(p-1)$.

Working:

If $\displaystyle v_1,...,v_n$ is a basis of $\displaystyle V$ then there is an invertible matrix with these vectors as columns.

Note that $\displaystyle v_1$ can be any non-zero vector, then $\displaystyle v_2$ can be any vector in $\displaystyle V$ that is not on the line spanned by $\displaystyle v_1$, and same for $\displaystyle v_3$ to $\displaystyle v_n$. So there are $\displaystyle p^n-1$ choices for $\displaystyle v_1$, $\displaystyle p^n-p$ choices for $\displaystyle v_2$, $\displaystyle p^n-p^2$ choices for $\displaystyle v_3$, etc.

So the order of $\displaystyle GL_n (F_p)$ is $\displaystyle \prod_{i=0}^{n-1} (p^n-p^i)$

So this gives me the answer to part b, but how do I work out part a and c?

Thanks for any help you provide.