I am just reading the proof of a theorem about
$M(q^2)$=$PSL_2(q^2)\bigcup \left\{f|f(z)=\frac{az^q+b}{cz^q+d}, 0≠ad-bc≠k^2; a,b,c,d\in GF(q^2); z\in GF(q^2)\cup\left\{\infty\right\} \right \}$
telling that:
The groups $M(q^2)$ and $PGL_2(q^2)$, wherein $q$ is an odd number, are not isomorphic.
Somewhere in the middle of the proof, it is assumed that the elements of $GF(q)$ are squares in $GF(q^2)$. Any sparks for solving this step of the Theorem. Thanks.