I am trying to show that $PSL_2(F_5)$ is isomorphic to $A_5$. I have shown that $|PSL_2(F_5)=60|$ I have found 2 matrices
$$ A= \left[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right] $$ $$B= \left[ \begin{array}{cc} 0 & -1 \\ 1 & 1 \end{array} \right]$$
such that $A^2=B^3=(AB)^5= I_2$. Thus there is a homomorphism from $A_5 \to PSL_2(F_5)$. I want to show that $A$ and $B$ generate $PSL_2(F_5)$ which would imply that the homomorphism is an isomorphism.
Any help is appreciated.
Your homomorphisms is injective because its domain is a simple group. It follows from that that your two elements generate a subgroup of order 60 which is therefore the whole group.