I am reading Apostol's Modular Functions and Dirichlet Series in Number Theory. I am confused over the definition of functions automorphic under subgroup $\Gamma_0(p)$. On page 79 of Section 4.4, Apostol defineds this by "$f(V\tau)=f(\tau)$ for every transformation $V$ in $\Gamma_0(p)$". But in the definition of modular functions, $f(A\tau)=f(\tau)$ for every transformation $A$ in the modular group $\Gamma$.
My question is, $\Gamma_0(p)\leqslant\Gamma$, are all modular functions automorphic under $\Gamma_0(p)$ by this definition?
Yes. But there will also be some functions which are invariant under the smaller subgroup $\Gamma_0(p)$ but not under the full modular group $\Gamma$.