This question refers to the top of p.156 in Koblitz's "Introduction to Elliptic Curves and Modular Forms," as well as to the beginning of the proof of Proposition 32a) at the bottom of that same page. This is self-study.
We take a lattice $L \subset \mathbb C$ and we let $t \in \mathbb C/L$ be a point of exact order $N$ mod $L$. We then consider all larger lattices $L'$ in which $L$ is of index $n$ such that $t$ is of exact order not only mod $L$, but mod $L'$. I view this as a denser lattice $L'$ properly containing $L$ and exactly $n - 1$ other translates of $L$.
The author says the latter of the two conditions is the same as requiring that "the only multiples $\mathbb Zt$ which are in $L'/L$ are the multiples $\mathbb ZNt$ which are in $L$."
Question 1: What does this mean, and why is it equivalent to the original wording of the condition? Every interpretation I come up with does not make any sense logically. What is a multiple $\mathbb Zt$, for starters?
The author then goes on to note that any such $L'$ corresponds to a subgroup of order $n$ in $\frac{1}{n}L/L$ (the subgroup being $L'/L$, I believe). This is fine, but he then makes two perplexing statements at the bottom of this same page during the proof of Proposition 32 a).
First, he says that if $(m, n) = 1$, then, working with $mn$ instead of $n$ under the above setup, the corresponding $L'$ not only correspond to subgroups $S'$ of order $mn$ in $\frac{1}{mn}L/L$ (as we noted above), but they correspond to the subgroups intersecting the subgroup $\mathbb Zt \subset \mathbb C$ trivially.
Question 2: Why do the subgroups have to intersect $\mathbb Zt$ trivially?
Finally, still on the same page, the author takes such a subgroup $S'$ and uses $(m, n) = 1$ to somehow find a unique subgroup $S''$ order $n$, and claims it has nontrivial intersection with $\mathbb Zt$.
Question 3: Where does this unique subgroup $S''$ come from, and how can a subgroup of a group that is supposed to have trivial intersection with $\mathbb Zt$ somehow have nontrivial intersection with $\mathbb Zt$?