I am intersted in Exercise 7 from Chapter 5 of this notes:
"Let $V$ be the set of non-zero lattices $L \subset \mathbf{C}$ that satisfy $x \cdot L \subset L$ for every $x \in \mathbf{Z}\left[\frac{1+\sqrt{-23}}{2}\right]$. Show that $V$ becomes an abelian group if we set $L_1 \cdot L_2=\{\sum x_i y_i \in \mathbf{C}: x_i \in L_1, y_i \in L_2\}$. Let $P \subset V$ be the set of lattices of the form $L_z=\mathbf{Z} \cdot z+\mathbf{Z} \cdot \frac{(1+\sqrt{-23}) z}{2}$ with $z \in \mathbf{C}^*$. Show that $P$ is a subgroup of $V$ and that $V / P \cong \mathbf{Z} / 3 \mathbf{Z}$."
I am not sure about the inverse of a lattice $L \in V$ with respect to the binary operation defined in the statement. It seems to me that each lattice in the set $V$ will correspond one-to-one to a fractional ideal of $\mathcal{O}_K=\mathbf{Z}\left[\frac{1+\sqrt{-23}}{2}\right]$ (Note that $\mathcal{O}_K$ is in fact the ring of integers of $K=\mathbb{Q}(\sqrt{-23})$). Therefore I think that the inverse of $L \in V$ might be $$L'=\{x \in \mathbf{C}: xL \subseteq L\}.$$ However I don't know how to verify that $L'$ is indeed a lattice and, furthermore, lives in $V$. Or if I should modify my $L'$.
I have already checked that $\mathcal{O}_K$ is itself a lattice that plays a role as an identity element in $V$.
Any help will be appreciated