I have prove that $PSL(2, 5)$ is isomorphic to $A_5$.
I understand, that I should do it with Sylow subgroups. There are 5 2-subgroups in $PSL(2, 5)$, but I can't describe them.
After that I can use group action on $PSL(2, 5)$ (conjugation with Sylow subgroups) and get homomorphism to $S_5$. Image of this homomorphism is our $A_5$.
So I have two problems with this prove : I can't describe Sylow subgroups of $PSL(2, 5)$ and don't understand how show that image of homomorphism is $A_5$.
I can't use the fact that $PSL(2, 5)$ is simple.