Generators of $SL_{2}(\mathbb{F}_p)$?

Question

Do the row operations that preserve the determinant generate the special linear group over the field with p elements (p is a prime)? In other words are the matrices: \begin{bmatrix}1&1\\0&1\end{bmatrix} \begin{bmatrix}1&0\\1&1\end{bmatrix} generators for $SL_{2}(\mathbb{F}_p)$?

Answer 1

Let $S=\pmatrix{1&1\\0&1}$ and $T=\pmatrix{1&0\\1&1}$. To show these generated $\text{SL}_2(\Bbb F_p)$, all you need to do is to prove that if $A\in \text{SL}_2(\Bbb F_p)$ then one can left-multiply $A$ by a sequence of $S$s and $T$s and arrive at the identity.

Let $A=\pmatrix{a&b\\c&d}$. Then multiplying on the left by $I$ or $T$ will give a matrix $\pmatrix{a_1&b_1\\c_1&d_1}$ with $c_1\ne0$. Now multiplying by a suitable power of $S$ gives a matrix $\pmatrix{1&b_2\\c_2&d_2}$. Multiplying by $T^{-c_2}$ gives $\pmatrix{1&b_3\\0&1}$. Finally multiply by $S^{-b_3}$ to get to the identity.

Answer 2

Yes, it's true. Gorenstein says this is due to Dickson. It's on p. 44 of Gorenstein's Finite Groups as Theorem 8.4

If you want as much detail as you like, both Gorenstein's Finite Groups and Hupperts Endliche Gruppen, Vol I contain extensive discussion of the structure of $SL(2,p)$ and related groups (in section 8 of chapter 2 in both cases).