I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995).
The exercise is: Prove that $$PSL(2,9)\cong A_6.$$
Let $G:=PSL(2,9)$, I can see that $|G|=360=2^3.3^2.5$ and it is enough to show that $n_5\ne 36$ (where $n_5$ be the number of Sylow $5-$subgroups) but I stuck here.
Thanks.