I'm reading through Diamond and Shurman's modular forms book. i'm having problems deciphering the meaning of some notation in section 4.2. For $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \textrm{SL}_2(\mathbb{Z})$ and $\tau \in \mathbb{H}$ denote $$j(\gamma,\tau) = c\tau + d.$$ Let $N \geq 1$ and $k \geq 3$ be integers and $\delta \in \textrm{SL}_2(\mathbb{Z})$. Let $P$ be the subgroup of $\textrm{SL}_2(\mathbb{Z})$ generated by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and let $\Gamma(N)$ be the principal congruence subgroup of level $N$. The book defines a function on $\mathbb{H}$ by $$E^{\delta}_k(\tau) = \sum_{\gamma \in (P \cap \Gamma(N)) \setminus \Gamma(N) \delta} j(\gamma,\tau)^{-k}.$$ The book does not explain the notation used to index the sum, but I took it to mean "sum over a set of coset representatives for $P \cap \Gamma(N)$ in $\Gamma(N)$ multiplied on the right by $\delta$". However, in deducing the transformation law for the above function under $\textrm{SL}_2(\mathbb{Z})$, they switch the order of $\Gamma(N)$ and $\delta$ to write $$j(\gamma,\tau)^{-k}E^{\delta}_k(\gamma\tau) = \sum_{\gamma' \in (P \cap \Gamma(N)) \setminus \delta\Gamma(N)} j(\gamma'\gamma,\tau)^{-k},$$ justifying the switch by the normality of $\Gamma(N)$ in $\textrm{SL}_2(\mathbb{Z})$. Here $\gamma\tau$ denotes the action of $\gamma$ on $\tau$ by linear fractional transformation. I can't seem to make sense of this notation. It looks like a set of coset representatives in $\delta\Gamma(N)$, but $\delta\Gamma(N)$ is not a group so this doesn't make any sense.
The end goal of the calculation is to show that $$j(\gamma,\tau)^{-k}E^{\delta}_k(\gamma\tau) = \sum_{\gamma'' \in (P \cap \Gamma(N)) \setminus \Gamma(N)\delta\gamma} j(\gamma'',\tau)^{-k}.$$ This seems clear however just by observing that $$\{\gamma'\gamma : \gamma' \in (P \cap \Gamma(N)) \setminus \Gamma(N)\delta\} = (P \cap \Gamma(N)) \setminus \Gamma(N)\delta\gamma.$$ I don't see why normality is needed.
If $\Gamma$ is normal in $SL_2(\Bbb{Z})$ for a fixed $\delta\in SL_2(\Bbb{Z})$ $$ \delta\Gamma=\Gamma \delta = \bigcup_l P g_l \qquad (disjoint\ union) $$ where $P = \Gamma \cap \langle p \rangle=\langle p^N \rangle$ with $p(z)=z+N$.
$$E_{\Gamma,\delta,k}(z)=\sum_l (Pg_l)'(z)^{k/2} \qquad is \ well-defined :\ \ (p^m g_l)'(z)=g_l'(z)$$
For all $\beta \in \Gamma $ $$\bigcup_l P g_l\beta=\delta\Gamma\beta=\delta\Gamma=\bigcup_l P g_l$$ and hence $$E_{\Gamma,\delta,k}(z)=\sum_l (Pg_l\beta)'(z)^{k/2}=\sum_l (Pg_j)'(\beta(z))^{k/2}\beta'(z)^{k/2}=E_{\Gamma,\delta,k}(\beta(z))\beta'(z)^{k/2}$$ which proves $$E_{\Gamma,\delta,k}\in M_k(\Gamma)$$