In the book Introduction to Elliptic Curves and Modular Forms from Neal Koblitz there is a certain step in the solution of an exercise that is left to the reader (IV.3, Problem 5 and solution). Since I failed to show this step I hope that someone here could help me.
First we define
$\Gamma_0(N) := \left\{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} \in SL_2(\mathbb{Z}) : c \equiv 0\mod N \right\}$
$\Gamma_1(N) := \left\{ \begin{pmatrix} a & b\\ c & d \end{pmatrix} \in \Gamma_0(N) : a \equiv d \equiv 1\mod N \right\}$.
Let $\gamma = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \in \Gamma_0(N)$ be such that $p^2 \mid b$. Let $\gamma_j \in \Gamma_1(N)$ and $\tau_j = \gamma^{-1}\gamma_j \gamma \in \Gamma_1(N)$.
Now comes the mentioned step:
Verify that if $\tau_{j}$ and $\tau_{j'}$ are in the same right coset of $\Gamma_1(N) \cap\alpha^{-1} \Gamma_1(N)\alpha$ in $\Gamma_1$ where $\alpha = \begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix}$ and $p>2$ prime then so are $\gamma_j=\gamma \tau_j\gamma^{-1}$ and $\gamma_{j'}=\gamma \tau_{j'}\gamma^{-1}$.
If I'm not mistaken, then I have to show:
Suppose $\tau_j = \beta \tau_{j'}$ for a $\beta \in \Gamma_1(N)$ then there exists $\tilde{\beta} \in \Gamma_1(N)$ such that $\gamma_j = \tilde{\beta}\gamma_{j'}$.
This seems like an easy problem in an algebra class, but I was not able to do prove that...
I'm only interested in the case $p|N$, if it helps you.
I appreciate any help.