Do the circle groups have any interesting stand-alone descriptions?

Question

By the circle groups, I mean firstly the circle group $\mathbb{T} \subseteq \mathbb{C}$ of all complex numbers having modulus $1$, and secondly the commutative group $\mathbb{S} = \mathbb{T} \cap \{x+iy \mid x,y \in \mathbb{Q}\}$ of all 'rational' such points.

Edit. I am also interested in the group $\mathbb{Q} / \mathbb{Z}.$

Question. Do the aforementioned "circle groups" have any interesting stand-alone descriptions, not referencing the usual number systems? For example, can we describe them as the initial objects of (or free objects in) some easily-motivated category of groups? Can we define them axiomatically, as the sole models of some second-order schema of axioms?

Thanks.

Answer 1

There is surely some phrasing of $\mathbf{T}$ as being the unique continuous group structure (up to isomorphism) on the unique compact one-dimensional manifold (up to isomorphism).

I doubt you can say anything nicer about $\mathbf{S}$ than it's the direct sum of the cyclic group on 4 elements with the free Abelian group on countably infinitely many generators. (I think that's a correct description)