From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn abstract signal theory from the book Unified Signal Theory by Cariolaro and I'm confused about several things. First, when discussing LCA groups, Cariolaro only mentions isomorphisms of groups, then says
For Topology, two isomorphic groups are essentially the same objects
and then states (sec 3.4):
If a group $G$ is lca and $G\sim H$ then $H$ is lca.
How can anything be said about the topology $H$ from a mere group-isomorphism? Did the author forget to say we're talking about continuous homomorphisms as arrows? Did he really mean "For topology two topologically isomorphic groups are essentially the same object"?
Just a bit later, the author states a theorem he attributes to Weil:
Every lca group $G$ has the form $G\sim \mathbb R ^p \oplus \mathbb Z ^q \oplus \mathbb O ^r$, where $\mathbb O$ is the trivial group.
Yet everywhere I looked I only saw that $G\sim \mathbb R ^p \oplus K \oplus D$ where $K$ is compact and $D$ is discrete. Why is the theorem as stated by the author true?
Added: Qiaochu Yuan pointed out the structure theorem as stated above is simply wrong. Under what conditions is it true? What kind of groups are isomorphic to $\mathbb O ^r$? I know that for compactly generated $G$ we have $G\sim \mathbb R^p \oplus \mathbb Z ^q \oplus K$ for compact $K$. Does this help?
The author is being first imprecise and then being wrong. For example, $\mathbb{R} \cong \mathbb{R}^n, n \ge 2$ as abstract groups, but not as topological groups. He should be and isn't talking about topological group isomorphisms. Second, you are of course correct that the classification of LCA groups is more complicated than that. For example, that classification doesn't include $S^1$, or nonzero finite abelian groups...
Edit: The correct statement is that every LCA subgroup of $\mathbb{R}^n$ can be written, after a suitable change of coordinates, in the form $\mathbb{R}^p \times \mathbb{Z}^q \times 1^r$ where $p + q + r = n$ and that last bit of notation (which makes no sense if the author is talking just about groups as opposed to subgroups; the author is not at all careful about making this distinction) has the following meaning: if $G_i$ is a sequence of subgroups of $\mathbb{R}$ then $G_1 \times G_2 \times \dots \times G_n$ is the corresponding subgroup of $\mathbb{R}^n$.
This claim is made more or less correctly earlier in the section but Theorem 3.3 is just clearly wrong as stated, even by the author's own admission: the author explicitly mentions the existence of finite cyclic groups earlier in the chapter. I don't know what's up with that.
A related warning: "$n$-dimensional LCA group" means "LCA subgroup of $\mathbb{R}^n$."
Edit #2: Theorem 3.6 makes a more general claim contradicting Theorem 3.3 and it is also wrong as stated. It correctly describes all LCA groups that arise by quotienting an LCA subgroup of $\mathbb{R}^n$ by another such subgroup but misses many other groups, e.g. on the one hand interesting discrete abelian groups such as $\mathbb{Q}$ and on the other hand interesting compact abelian groups such as the $p$-adics.