On p.239 A Course in Computational Number Theory, Cohen writes
"Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ defined in Section 5.2, we can define an operation between forms, which we call composition, which becomes a group law only at the level of classes modulo $\mathrm{PSL}_2(\mathbb{Z})$."
My question is, should the word "only" in boldface be omitted?
The domain of the map $\phi_{FI}$ Cohen just referenced is the set of integral binary quadratic forms of some fixed nonsquare discriminant modulo the action of the subgroup $\Gamma_{\infty}$ of $\mathrm{SL}_2(\mathbb{Z})$ given by
$$ \Gamma_{\infty} = \left\{ \, \begin{bmatrix} 1 & m \\ 0 & 1 \end{bmatrix}, m \in \mathbb{Z} \, \right\}. $$
Cohen defines composition by an explicit version of the method of Arndt and goes out of his way to say that the class of the resulting form modulo the action by $\Gamma_{\infty}$ is canonically determined. It is clear also that composition restricts to a well-defined operation on pairs of classes modulo $\Gamma_{\infty}$.
I would infer from these observations, and his use of the word only, that this operation on primitive classes modulo $\Gamma_{\infty}$ does not form these classes into a group. I would, that is, if I didn't already know that they did form a group. A reasonably detailed proof is in Section 8 of this paper of Lenstra.
Perhaps they don't form a group, and I am missing some assumption in Lenstra's paper? If so, maybe Cohen is taking issue with the primitivity of the forms. But I think not, since he says composition does give a group at the level of the class group, so it seems implicit he is talking about primitive classes. Am I missing something else? Perhaps he just made a poor choice of wording?