I'm trying to learn about group theory (I'm neofite in this field) for physics reading a book, "Physics from symmetry" by Schwichtenberg. In section 3.2 the author introduces rotations in two dimensions and talks about the $SO(2)$, then examines rotation using complex numbers and introduces the $U(1)$ group.
Then the author introduces rotation in 3 dimensions using matrices, thus introducing the group $SO(3)$, and quaternions, thus introducing the group $SU(2)$. Here the question is about the $SU(2)$ group. Reading online, I think it is correct to say that $SU(2)$ is a subgroup of $U(2)$, which is the group of 2x2 unitary matrices.
From this point of view the 2 makes sense, since quaternions can be represented using 2x2 matrices, but on the other hand, unit quaternions have 3 degrees (fixing the length) of freedom, not 2.
I understand that in order to avoid reflections it is necessary to define a special unitary group, but why for common complex numbers there is no such thing as a $SU(1)$ group? And moreover, why does the author chooses not to talk about the $U(2)$ group? Does he skips over it as it is not useful, or is there a mathematical reason for which $U(2)$ cannot be taken into consideration?
P.S.: Thank you in advance for any of your eventual answers, and excuse my poor English: I'm still learning it!