Prove that $PSL(2,9)\cong A_6.$

Question

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.

Answer 1

All we need to do is to construct a subgroup of index 6 in $PSL_2(\mathbb F_9)$ (indeed, such a subgroup gives a non-trivial homomorphism to $S_6$; since $PSL_2(\mathbb F_9)$ and $A_6$ are both simple of the same order, this is an isomorphism).

Now there is a beautiful geometric construction of a (non-trivial) map $A_5\hookrightarrow PSL_2(\mathbb F_9)$: $A_5$ is the group of rotations of the dodecahedron so we have a map $A_5\hookrightarrow SO(3)\cong PSU_2\subset PSL_2(\mathbb C)$; this map is defined over $\mathbb Z[i,\frac12,\phi]$ (where $\phi$ is the golden ratio), and reduction mod 3 gives a map $A_5\hookrightarrow PSL_2(\mathbb F_3[i]\cong\mathbb F_9)$. Voilà.

Answer 2

An easy argument is given in p.52, R.A.Wilson, "The finite simple groups", Graduate texts in mathematics, Springer-Verlag.