Being $F$ the field of order $q$ that linear groups are defined here, there is something I can't understand.
I know that $|SL(n,q)| = \dfrac{|GL(n,q)|}{(q-1)}$, and I know that $PSL(n,q) = SL(n,q)/Z(SL(n,q))$. But I'm having problems with something:
We have $Z(SL(n,q)) = \{\lambda I_d \; | \; \lambda \in F,\lambda^n=1\}$, so $|Z(SL(n,q))|$ is the number of solutions for the equation $\lambda^n=1$ in $F$. I found that the number of solutions is given by $\gcd(n,q-1)$, but why $\gcd(n,q-1)$?
Thanks.