Index and Equality Conditions of the Subgroups $\Gamma_{0,n}(N)$ and $\Lambda_n(N)$

Abstract

In this paper, we find conditions on the natural number $n$ that the subgroups $\Gamma_{0,n}(N)$ and $\Lambda_n(N)$ of modular group are different. And then, by defining an $\Lambda_n(N)$ invariant equivalence relation on the subset $\hat{\mathbb{Q}}_n(N)$, we calculate the index formula for $\Gamma_{0,n}(N)$ in $\Lambda_n(N)$.